remarkable inequalities based on convex and concave functions

1 Author : K.Santhanam

Remarkable inequalities based on the properties of convex and concave functions

Author : K.Santhanam, M.Sc.,M.Phil.

Abstract

The present text is aimed to a thorough introduction to convex, concave set and

function, which entails a powerful and elegant interaction between analysis and

geometry. Convex functions are powerful tools for proving a large class of inequalities.

They provide an elegant and unified treatment of the most important classical

inequalities. A general inequality is proved using the definition of convex functions.

Many major inequalities are deduced as applications.

The Introductory chapter attempts to trace the basic concept of convex set,

concave set and its properties.

The second chapter entitled Convex Functions and its properties with some useful

results and examples.

The third chapter entitled Important Inequalities with convex function and

Lebesque measure, which has some basic concepts of σ – algebra, Lebesque measure,

Lebesque integral

The fourth chapter entitled Concave Functions and its properties with some

interesting results and examples.

The fifth chapter entitled Some Special Inequalities which dealt with derivation of

special inequalities using the convex, concave functions.

The concluding chapter summarizes the argument and makes a list of findings and

future progressing.


1. Introduction

The study of convex sets is a branch of geometry, analysis, and linear algebra that

has numerous connections with other areas of mathematics and serves to unify many

apparently diverse mathematical phenomena. It is also relevant to several areas of science

and technology.

As per J. L. W. V. Jensen, It seems to me that the notion of convex function is just

as fundamental as positive function or increasing function. If am not mistaken in this, the

notion ought to find its place in elementary expositions of the theory of real functions.

Convex functions play an important role in many branches of mathematics

including probability and statistics, optimization problems in linear programming, as well

as other areas of science and engineering.

Convexity is a simple and natural notion which can be traced back to Archimedes

(circa 250 B.C.), in connection with his famous estimate of the value of π (using

inscribed and circumscribed regular polygons). He noticed the important fact that the

perimeter of a convex figure is smaller than the perimeter of any other convex figure,

surrounding it.

The first systematic study of convexity was made by Minkowski (1864- 1909),

whose works contain, most of the important ideas of the subject. The early developments

of convexity theory were finite-dimensional and directed mainly toward the solution of

quantitative problems; an excellent survey of them was made by Bonnesen and Fenchel

in 1934.


Definition: 1.1(Convex combination)

Let X be any set , the convex combination of the set X is defined as follows

For any with 0 < λ < 1, the combination ( ) is called the convex

combination of x, y.

In general , If , then convex combination of these points is

given by where , 0 <

Definition: 1.2 (Convex Set)

A convex set is the set which contains all possible convex combinations of its points.

(i.e) A convex set is a set of all points such that the line joining any two points of that set

within the set.

(i.e) If a set X is said to be convex , then for any x,y ϵ X with 0 < λ <1 such that

λx +(1-λ)y ϵ X

In Euclidean space, a convex set is the region such that, for every pair of points within

the region, every point on the straight line segment that joins the pair of points is also

within the region.

For example, a solid cube is a convex set and a crescent shape, is not convex.

Geometrical interpretation of convex and non-convex sets:

https://en.wikipedia.org/wiki/Euclidean_space

https://en.wikipedia.org/wiki/Region_%28mathematics%29

https://en.wikipedia.org/wiki/Straight_line

https://en.wikipedia.org/wiki/Cube_%28geometry%29

https://en.wikipedia.org/wiki/Crescent


Convex sets Non-Convex sets

Definition: 1.3 (Concave set)

A set which is not convex is called non-convex set or concave set

Theorem 1.1

Prove that the any closed interval [a,b] in R is a convex set.

Proof

, -

( ) , -

(i.e) to prove ( )

Since , -,

⟹

⟹ (∵λ > 0) ------- (1)

and ( ) ( ) ( ) (∵ (1-λ) > 0) -------- (2)

(1) + (2) ⟹ +( ) ( ) ( )

⟹ +a – ( )


⟹ a ( )

⟹ ( ) , -

∴ the interval [a,b] in R is a convex set.

Theorem 1.2

Prove that S = * ∶ + is a convex set

Proof.

Let S and 0 < λ < 1 , we have to prove ( ) S

Since S,

⟹ and -------- (1)

To prove ( ) S, it is enough to prove

( ( ) ) ( ( ) )

Now ( ( ) ) ( ) ( ) ,( by using (1) )

⟹ ( ( ) )

⟹ ( ( ) ) ------- (2)

Now ( ( ) ) ( ) ( ) ,( by using (1) )

⟹ ( ( ) )

⟹ ( ( ) ) ------ (3)

From (2) & (3), S = * ∶ + is a convex set

Theorem 1.3

If S and T are two convex sets in then S ⋂ T is also convex set

Proof.

Let x, y ϵ S ⋂ T, Choose 0 < λ < 1

To prove S ⋂ T, it is enough to prove ( ) S ⋂ T


For all x, y ϵ S ⋂ T, we have x, y ϵ S and x, y ϵ T

Since S and T are convex, ⟹ ( ) and ( )

⟹ ( ) S ⋂ T

⟹ S ⋂ T is convex set

Note : 1.1

Union of two convex sets need not be a convex set

For example consider the closed intervals [0,1] and [2,3]

By Theorem [0,1] and [2,3] are convex set

Consider A = [0,1] U [2,3]

Clearly 1,2 ϵ A ( ∵ 1 ϵ [0,1] and 2 ϵ [2,3])

Choose 0 < λ < 1, then ( )

Clearly ∉ A for all λ such that 0 < λ < 1

(∵ If we take

, then

which is not in both [0,1] and [2,3]

Hence ∉ A)

⟹ Union of two convex sets need not be a convex set

Theorem 1.4

If S is convex set, then T = { x: Ax+b = y , y ϵ S } is also convex.

Proof

Let ϵ T and 0 < λ < 1

We have to prove T is convex set ,

(i.e) to prove ( )

(i.e) to prove, there exist a y ϵ S such that ( ( ) )


Since ϵ T, there exist such that nd ----(1)

Now ( ( ) ) = ( )

= ( )

= ( )

= ( ) ( )

= ( ) ( )( )

= ( ) ( by using (1) )

Since and S is a convex set, hence ( )

∴ ( ( ) ) ( )

Let ( )

⟹ ( ( ) ) , where y

⟹ ( )

⟹ T is a convex set.


2. Convex Functions and its properties

Definition 2.1.1: (Convex function)

A real-valued function φ(x) defined on an interval is called convex (or convex downward

or concave upward) if the line segment between any two points on the graph of the

function lies above or on the graph.

(i.e) A real function φ defined on a segment (a, b), where -∞ ≤ a < b ≤ ∞ is called convex

if the inequality (( ) ) ( ) ( ) ( ) holds whenever a < x < b,

a < y < b, and 0< λ< 1.

Definition 2.1.2 (Convex function)

Let φ be a real function which is defined on a real interval I.

( ) ( ) ( )

Examples.2.1.1


The quadratic function ( ) for any real number x is convex.

Solution

To prove φ is convex, (i.e) to prove ( ( ) ) ( ) ( ) ( ) for

any λ such that and any real x,y

(i.e) to prove ( ( ) ) ( )

(i.e) to prove ( ( ) ) ( ( ) )

Now ( ( ) ) ( ( ) )

( ) ( ) ( )

( ) (( ) ( )) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )( )

( )( )

Clearly ( ) ≥ 0 for all real x,y

Since

∴

Hence ( ( ) ) ( ( ) ) ( )( )

∴ ( ) is a convex function

Example 2.1.2.

The absolute value function ( ) for any real number x is convex.


Solution

To prove φ is convex, (i.e) to prove ( ( ) ) ( ) ( ) ( ) for any

λ such that and any real x,y

Now ( ( ) ) = ( )

≤ ( ) (By using triangular inequality |a + b| ≤ |a| + |b|)

( )

( ) ( ) ( )

Hence ( ) is a convex function

Lemma 2.1.1


∶ ( ) ( )

( ) ( )

Proof.

Suppose φ is convex,

Let

Now

⟹ ( ) .

/

Let λ =

Then

( )

∴ ( ) .

/ ( ( ) )

( ) ( ) ( ) ( By using convexity of φ)

⟹ ( )

( )

( )


Subtract both sides by ( ),

( ) ( )

( )

( ) ( )

Divide both sides by ( )

( ) ( )

( )

( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( )

( )

( ) ( )

( )

( )

( )

( )

( )

( )

( )

( ) ( )

( )

∴ ( ) ( )

( ) ( )

( )

Converse part is similar to the reverse of the above proof.

Lemma 2.1.2


∶ ( ) ( )

( ) ( )

Proof.


Let

Now


Let λ =

Then

( )

⟹ ( )

( ) ( ( ) ) ( ) ( ) ( ) ( by convexity of φ )

Let ( ) ( ) ( )

∴ ( ) ------- (1)

Let us take ( ) ( ) ( )

, the slop of the chord joining the points

( ( )) . ( )/

( ) ( ) ( )

( )

, (∵ from (1) ( )) -------(2)

But ( )

( ) -------- (3)

Also ( ) ( )

( ) ( )

( ) ( By using (1) ) -----(4)

From (2),(3) and (4)

( ) ( )

( ) ( )

Geometrical interpretation:

The slope of is less than or equal to that



Definition 2.2.1 (Strictly Convex function)

If the inequality above is strict for all x and y, then φ(x) is called strictly convex.

(i.e) A real function φ defined on a segment (a, b), where -∞ ≤ a < b ≤ ∞ is called convex

if the inequality (( ) ) ( ) ( ) ( ) holds whenever a < x < b,

a < y < b, and 0< λ< 1.

Definition 2.2.2 (Strictly Convex function)


( ) ( ) ( )

Lemma 2.1.3


∶ ( ) ( )

( ) ( )


Proof.

Similar to the proof of Lemma 2.1.1

Lemma 2.1.4


∶ ( ) ( )

( ) ( )

Proof.

Similar to the proof of Lemma 2.1.2

Example 2.2.1.

The quadratic function ( ) for any real number x is strictly convex.

Solution

As per the Example 2.1.1, we have

( ( ) ) ( ( ) ) ( )( )

( )( ( ) ) ( ( ) )

( )( ( ) ) ( ( ) )

( ) ( ( ) ) ( ) ( ) ( )

Hence ( ) for any real number x is strictly convex

Lemma 2.1.5

A real differentiable function φ is convex in (a, b) if and only if a < s < t < b implies

φ’(s) ≤ φ’(t)

Proof.

If x < t < y, then the point (t, φ(t)) should lie below or on the line connecting the points

(x, φ(x)) and (y, φ(y)) in the plane.


Suppose φ is a real differentiable convex function in (a, b) and a < s < t < u < b

By definition of a differentiable function, φ’(s) = ( ) ( )

And , φ’(t) = ( ) ( )

Given s < t , let x = αs + βt where β= 1 – α --------- (1)

Since φ is convex ( ) ( ) ( ) --------- (2)

(1) ⟹ ( – ) – ( )

∴ ( )

⟹

Now – –

( )

( )

(2)⟹ ( ) .

/ ( ) .

/ ( )

( ) ( ) .

/ ( ) .

/ ( )

Multiply L.H.S of above by ( ) ( )

, we have

( ) ( )

( ) .

/ ( ) .

/ ( )

( ) .

/ ( ) .

/ ( ) .

/ ( ) .

/ ( )

⟹ .

/ ( ( ) ( )) .

/ ( ( ) ( ))

⟹ ( ( ) ( ))

( ( ) ( ))

Given φ is differentiable, let x approaches ‘s’ on the right

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

----------- (3)


let x approaches ‘t’ on the left

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

---------- (4)

From (3) and (4) , ( ) ( )


Lemma 2.1.6

A real differentiable function φ is convex in (a, b) if and only if the derivative φ’ is a

monotonically increasing function.

Proof:

As per the Lemma 2.1.5, ( ) ( ) for all s < t

Hence φ’ is monotonically increasing function

If the inequality is strict then φ is strictly convex

( ) ( )


Example 2.2.2

The exponential function ( ) is strictly convex for all real x

Solution:

Let x, y be any real numbers such that

Given ( )

⟹ ( ) for all x

Clearly for all x < y

⟹ ( ) ( ) for all x < y


⟹ ( ) is strictly convex (By Lemma 2.1.6)

Lemma 2.1.7

A real differentiable function φ is convex in (a, b) if and only if the second derivative is

positive (i.e) φ’’ ≥ 0.

Proof:

Using Taylor series

( ) ( ) ( )( ) ( )( )

(Approximately)

Suppose second derivative of φ is positive, (ie) ( )

Clearly ( )( )

Hence ( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( ) ---------- (1)

Put t = x in (1)

( ) ( ) ( )( ) ------- (2)

Put ( ) in the last term of (2)

( ) ( ) ( )( ( ( ) ) )

( ) ( )( ( ) )

( ) ( )(( ) ( ) )

( ) ( ) ( )(( )( ) ) ---------- (3)

Put t = y in (1)

( ) ( ) ( )( ) ------- (4)


( ) ( ) ( )( ( ( ) ))


( ) ( )( ( ) )

( ) ( )( )

( ) ( ) ( )( ( )) ------------- (5)

Now ( ) ( ) ( )

⟹ ( ) ( ) ( )

{ ( ) ( )(( )( ))} ( ){ ( ) ( )( ( ))+

( ) ( )(( )( )) ( ) ( ) ( ) ( )( ( ))

( ) ( )( ( )( )) ( ) ( ) ( )( ( )( ))

( )

∴ ( ) ≤ ( ) ( ) ( )

Put ( )

Hence ( ( ) ) ≤ ( ) ( ) ( )

⟹ φ is convex


Definition 2.3.1 (Strongly Convex function)

If φ(x) has a second derivative in (a,b), then a necessary and sufficient condition for it to

be strongly convex on that interval is that the second derivative φ’’(x) > 0 for all x in

(a,b). And if φ’’(x) = m > 0 , then m is called strong convexity constant.

Note 2.3.1.

Clearly if φ(x) is strongly convex ⟹ φ(x) is strightly convex ⟹ φ(x) is convex.

But converse may not be true.


Example 2.3.1.

The quadratic function ( ) for any real number x is strongly convex.

Solution

Given ( )

⟹ φ’(x) = 2x

⟹φ’’(x) = 2 > 0 for all real x

Hence φ(x) is strongly convex and 2 is called strong convexity constant.

Example 2.3.2.

The function ( ) for any real number x, is strictly convex but not strongly

convex

Solution

Let

Now ( ( ) ) ( ( ) ) = (( ( ) ) )

< ( ( ) ) (by using Example 2.2.1)

( )

( ) ( ) ( )

Hence ( ( ) ) ( ) ( ) ( )

∴ φ(x) is strictly convex ⟹ φ(x) is convex

But ( )

⟹ ( ) for all real x

Hence ( )

⟹ ( )

∴ φ(x) is not strongly convex


Example 2.3.3.

The function ( ) for any real number x, is convex but not strictly convex

Solution

From Example 2.1.2, ( ) is convex

But ( ( ) ) ( ) ( ) ( )

Hence ( ) is not strictly convex.

Example 2.3.4.

The function ( ) for any real number x, is strictly convex but not strongly

convex

Solution

For any real x, y with x < y,

( ) then ( ) for all real x

Hence for all x < y , ( ) ( )

Hence ( ) is monotonically increasing

By Lemma 2.1.2, ( ) is strictly convex

But

Hence ( ) , for all real x

∴ ( ) is not strongly convex.

Example 2.3.4.

The function ( ) √ for any real number x, is not convex but monotonically

increasing.

Solution

Given ( ) √


⟹ ( )

⟹ ( )

, for all real x

Hence by Lemma 1.3 , φ(x) is not convex

But ( ) √ √ ( ) for all x < y

⟹ φ(x) is monotonically increasing

Example 2.3.5.

The function ( ) for any real number x, is strongly convex but not monotonically

increasing.

Solution

By Example 3.1, ( ) is strongly convex

Suppose -x < -y , then ( ) ( )

Hence ( ) is not monotonically increasing.

Note 2.3.2

If f is a function defined on the interval (a, b) which is differentiable, the four derivatives

are defined as follows, for any ( )

( )

( ) ( )

=

( ) ( )

(upper right derivative)

( )

( ) ( )

=

( ) ( )

(lower right derivative0

( )

( ) ( )

=

( ) ( )

(upper left derivative)

( )

( ) ( )

=

( ) ( )

(lower left derivative0


Note 2.3.3

If a function f defined on the interval (a,b) is continuous, then for any t ϵ (a,b)

( )

( )

( )

Lemma 2.1.8.

Let I be an open interval. If φ : I→ R is convex function (strictly convex), then the left

derivative ( ) and the right derivative (x) exist and are increasing (strictly

increasing) on I.

Proof.

We prove first that the right derivative exists.

Consider the function ( ) ( ) ( )

-------- (1)

Let ,

∵ , by definition 1.3of convexity , ( ) ( )

( ) ( )

⟹ ( ) ( )

( ) ( )

---- (2)

From (1) and (2) , ( ) ( ) for any

⟹ f is strictly monotonically increasing function

We have

( ) ( )

=

( ) ( )

is either finite or -∞

From (2) , ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

Hence

( ) ( )

=

( ) ( )

( ) ( )

⟹

( ) ( )

≠ - ∞


Hence

( ) ( )

is finite

⟹ ( ) is exist (lower right derivative)

And for all , ( ) ( )

( ) ( )

Taking limit as h → on both sides, we have ( ) ( )

Hence ( ) is increasing

Now we prove that the left derivative exists

Consider the function ( ) ( ) ( )

-------- (3)

Let ⟹

∵ , by definition 1.3of convexity ,

( ) ( )

( )

( ) ( )

( ) --------(4)

From (3) and (4) , ( ) ( ) for all

⟹ f is strictly monotonically decreasing function

We have

( ) ( )

=

( ) ( )

is either finite or +∞

From (4), ( ) ( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( ) ( )

( )

Hence

( ) ( )

=

( ) ( )

( ) ( )

( )

⟹

( ) ( )

Hence

( ) ( )

is finite

⟹ ( ) is exist (lower left derivative)


And for all , , where

⟹ ( ) ( )

( ) ( )

( by definition of convexity)

⟹ ( )

( )

⟹ Hence ( ) is increasing

Lemma 2.1.9

A convex function φ defined on an interval I is continuous

Proof.

Let a , we have to prove φ is continuous on I

(i.e) enough to prove ( ) ( )

( )

We have

( ) 0

( ) ( )

( ) ( ) ( )1

( ) ( )

( ) ( )

( )

( ) ( ) ( ) (∵ φ is convex and

( ) as per Lemma 2.1.4)

( ) ( ) ( ) = ( )

∴

( ) ( ) ----- (1)

We have

( ) 0

( ) ( )

( ) ( ) ( )1

( ) ( ) ( ) (∵ φ is convex and ( ) as per Lemma 2.1.4)

( ) ( ) ( ) = ( )

∴

( ) ( ) ------- (2)

From (1) and (2) , ( ) ( )

( )

Hence φ is continuous.


Note: 2.3.4

If a real function f(x) ≤ A for all x in X, then the function f is said to be bounded above by

A. On the other hand, if f(x) ≥ B for all x in X, then the function f is said to be bounded

below by B.

A real function f(x) is said to be bounded, if it is both bounded above and below.

Lemma 2.1.10

A convex function φ defined on the closed interval [a, b] is bounded.

Proof.

First to prove φ is bounded above:

Let M = max {φ(a),φ(b)} and let 0 < λ < 1

Since φ is convex, ( ( ) ) ( ) ( ) ( ) ---------- (1)

Since M = max {φ(a),φ(b)} , we have ( ) ( )

From (1), ( ( ) ) ( )

≤ M

Let ( )

Hence ( ) for all x in [a,b]

Hence φ is bounded above --------- (A)

Second to prove φ is bounded below:

Let x, y ϵ [a.b] and 0 < λ < 1

Since φ is convex, ( ( ) ) ( ) ( ) ( )

Choose λ =


.

/

( )

( )

⟹ ( )

.

/

( )

⟹ ( ) .

/ ( ) --------- (2)

Since M = max {φ(a),φ(b)} and y ϵ [a, b], we have φ(y) ≤ M

⟹ ( )

From (2), ( ) .

/

Let us take 2 .

/

∴ ( ) ( )

⟹ ( ) for any x ϵ [a,b]

⟹ φ is bounded below ------- (B)

From (A) and (B) , φ is bounded.

Lemma 2.1.11

If φ and ψ are any to convex functions defined on the interval I, then any linear

combination of is also convex provided α, β are non-negative real numbers.

Proof.

Let ( ) ( ) ( )

Let x, y ϵ I and 0 < λ < 1

Now ( ( ) ) ( ( ) ) ( ( ) )

By using convexity of φ and ψ ,

( ( ) ) , ( ) ( ) ( )- , ( ) ( ) ( )-

, ( )- ,( ) ( )- , ( )- ,( ) ( )-


, ( ) ( )- ( ), ( ) ( )-

( ) ( ) ( )

⟹ ( ) ( ) ( ) is convex.

Lemma 2.1.12

If φ and ψ are any to convex functions and ψ is increasing on I, then the composition

ψ ◦ φ is also convex on I

Proof.

Let x, y ϵ I and 0 < λ< 1

Let ( ) = ( ) ( ( ))

Now ( ( ) ) ( ( ( ) )) ---------- (1)

Since φ is convex, ( ( ) ) ( ) ( ) ( )

From (1), ( ( ) ) ( ( ) ( ) ( ))

( ( )) (( ) ( )) ----------- (2)

Since ψ is convex, ( ( )) (( ) ( )) ( ( )) ( ) ( ( )

From (2), ( ( ) ) ( ( )) ( ) ( ( ))

( ) ( ) ( )

∴ ( ( ) ) ( ) ( ) ( )

⟹ ( ) = ( ) ( ( )) is convex.

Definition 2.4.1 (Dominated point)

A point P in the two-dimensional plane is said to dominate another point Q,

if their x – and y –coordinates satisfy the conditions: x(P) > x(Q) and y(P) > y(Q)

( ) If ( ) ( ) then


Example 2.4.1

The point (5,7) is dominate the point ( 3,1)

Definition 2.4.2 (Maximal Point)

A maximal point in a collection C of n points is a point which is not

dominated by any other point in the collection.

( ) If ( ) ( ) ( ), then the maximal point of these collection is

a point ( ) such that where

Example 2.4.2

Consider the collection C of points { (9,5) , (4,3), (8,1), (-2,-3) , (8,-5), (1,2), (7,4) }

The point (9,5) is the maximal point of the set C.

Lemma 2.1.13

Let φ : (a,b) → R be a continuous function. Then φ is convex if and only if

∫ ( )

, ( ) ( )- for all

Proof.


We have to prove

∫ ( )

, ( ) ( )- for all

We have to evaluate

∫ ( )

Put ( ) where 0 < λ < 1

⟹ – ( )

Limits :

When x = s , s = ( )

⟹ ( )


⟹ λ = 1

When x = t, t = ( )

⟹( )

⟹ λ = 0

∴

∫ ( )

∫

( ( ) )( )

∫

( ( ) )* ( )+

( )

∫

( ( ) )

∫

( ( ) ) ----------(1)

∫

( ( ) ) (by using the properties of definite integral)

Since φ is convex, ( ( ) ) ( ) ( ) ( )

(1) ⟹

∫ ( )

∫ , ( ) ( ) ( )-

∫ ( ) ∫ ( ) ( )

( ) ∫ ( ) ∫ ( )

( )

1

( ) {

1

}

( ) (

) ( )(

)

( ) (

) ( ) .

/

( ) ( )

∴

∫ ( )

( ) ( )

, for all

Conversely suppose

∫ ( )

( ) ( )

for all


We have to prove φ is convex

Suppose not, (ie) φ is not convex, then there exist s < t such that

( ( ) ) ( ) ( ) ( ), where 0 < λ < 1

Let us choose λ =

Hence we will get .

/

( )

( )

------ (2)

Consider the set 2 ( ) ∶ ( ) ( ) ( ) ( )

( )3

(2) ⟹ .

/

( )

( )

( ) ( ) ( )

( ) . ( ) ( )

/

( )

( ) . ( ) ( )

/

( )

> ( ) ( ) ( )

.

/

Hence .

/ ( )

( ) ( )

.

/ and

( )

∴

⟹ C is non-empty set

Clearly C is open

Let ( ) be the maximal point of the set C which contains


Since ( ( )) and ( ( )) are on the graph of the function

( ) ( ) ( ) ( )

( )

⟹ ( ) ( ) ( ) ( )

( ) --------- (2)

And ( ) ( ) ( ) ( )

( ) ----------- (3)

( )

( )

⟹

( ) ( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( ) ( ) ( )

0

1

∴ ( ) ( )

( )

( ) ( )

0

1 ------------ (4)

∴

∫ ( )

∫ 0 ( )

( ) ( )

( )1

,for all

∫ ( )

∫

( ) ( )

( )

( )

∫

( ) ( )

∫ ( )

( )

-

( ) ( )

(

)1

( )

( )

( ) ( )

2 .

/ .

/3


φ(s) ( ) ( )

2 .

/ ( )3

φ(s) ( ) ( )

2

( ) ( )3

φ(s) ( ) ( )

2

( )3

( ) ( )

( )

φ(s) ( ) ( )

2

( )( )3

( ) ( )

φ(s) ( ) ( )

2

( )3

( ) ( )

φ(s) ( ) ( )

.

/

( ) ( )

( by using (4) )

∴

∫ ( )

( ) ( )

which is contradiction to our hypothesis

∫ ( )

( ) ( )

Hence φ is convex.

Lemma 2.1.14.

The function φ is convex in the interval I if and only if the determinant |

( ) ( ) ( )

|

is non-negative for any in the interval I

Proof.

Suppose the determinant |

( ) ( ) ( )

| is non-negative for any in the

interval I


(i.e)|

( ) ( ) ( )

| ≥ 0

⟹ ( ( ) ( )) ( ( ) ( )) ( )( )

⟹ ( ) ( ) ( ) ( ) ( ) ( )

⟹ ( ) ( ) ( ) ( ) ( ) ( )

⟹ ( ) ( ) ( ) ( ) ( ) ( )

⟹ ( ) ( ) ( )

( ) ( )

------- (1)

Since , we can write

Let λ =

Then

( )

∴ ( ) , where 0 < λ < 1

( ) ( ( ) ) ( ) ( ) ( ) ( by using (1) )

⟹ φ is convex



3. Important Inequalities with convex function and Lebesque measure

Theorem 3.1.1. (Discrete-Jensen's Inequality).

Let φ be a convex function on the open interval I and let ( ). If

∑ , then (∑

) ∑ (

)

Proof

In order to prove this theorem, we use mathematical induction

Let , Consider

( ) ( ) ( ) ( by definition of convexity) -------- (1)

Hence the theorem is true for

Assume this theorem is true for n =

( ) (∑ ) ∑ (

) ----------- (2)

To prove this theorem is true for n = k

∑ ( ) = ( ) ∑

( )

(

)

( ) ( )∑ (

) where

( ) ( ) (∑ ) (By using (2))

≥ ( ( )∑ ) (by using (1) )

≥ . ( )∑

/

≥ . ( )

∑

/

≥ ( ∑ )

≥ (∑ )

⟹∑ ( ) ≥ (∑

)


Hence the theorem is true for n = k

By mathematical induction this theorem is true for all n

Definition 3.5.1. (σ – algebra)

Let X be a non-empty set. Let P(X) be its power set. Then a subset F of P(X) is called

a σ – algebra if it satisfies the following properties

1) F is non-empty. There is at least one subset A of X in F.

2) F is closed under complementation. If A is in F then Ac is also in F

3) F is closed under countable union. If A1, A2… are in F, then ⋃ in F

Note 3.5.1:

σ – algebra is closed for countable intersection.

Example 3.5.1.

On any non-empty set X, the power set P(X) is a σ – algebra. It is called discrete σ –

algebra

Example 3.5.2.

On any non-empty set X, the set {φ,X}is a σ – algebra. It is called minimal or trivial σ –

algebra.

Note 3.5.2:

Any σ – algebra of a non-empty set X, lies between {φ,X} and P(X)

Definition 3.6.1. (Measurable Space)

Let X be a non-empty set. By a σ – algebra M on X, We mean that a non-empty

collection of subsets of X such that

1) X M

2) If A M, then Ac M (M is closed for complementation)


3) If An M where n N, then ⋃ M (M is closed for countable union)

The pair (X,M) is called Measurable Space and the members of M are called measurable

subsets of X.

Example 3.6.1

Let X be any set. Let M consist only the empty set φ, then M is measurable space.

Example 3.6.2

Let X be any non-empty set. Let M consist of all subsets of X, then M is a measurable

space.

Example 3.6.3

Let X be any non-empty set. Let M consist of all subsets of X that are countable (finite),

then M is a measurable space.

Definition 3.7.1. (Measure)

A positive measure μ is a non-negative extended real valued set function defined

on a σ- algebra M such that μ(⋃ ) = ∑ ( )

whenever An∩ Am = ϕ , n ≠ m.

To avoid trivialities we shall assume that μ(A) < ∞ for some A ∊ M.

If the range of μ is in R, then μ is real. If the range of μ is in C, then μ is complex.

Example 3.7.1

Let X be any non-empty set. μ= P(X) ( Power set of X)

μ (A) = {

( denotes Cardinality of A = number of elements in A)

Clearly μ is non-trivial, that is ∃ A ∊ X such that μ (A) <∞

Let {An} be a sequence of pairwise disjoint measurable sets


If An‘s are infinite, then μ(An) = ∞ n

⇒ ∑ ( ) = ∞

Since each An is infinite, ⋃ = ∞

⇒ μ(⋃ ) = ∞

Hence μ(⋃ ) = ∑ ( )

If An’s are finite and having n elements each, then μ(An) = n n

⇒ ∑ ( ) = ∑

= ∞

Since each An having n elements, ⋃ = ∞

⇒ μ(⋃ ) = ∞

Hence μ(⋃ ) = ∑ ( )

Therefore μ is a measure and it is called the counting measure.

Example 3.7.2

Let X be any non-empty set and let x0 be a fixed point of X.

Define μx0(A) = 2 ∊

Clearly μ is non-trivial, that is ∃ A ∊ X such that μ (A) <∞

Let {An} be a sequence of pairwise disjoint measurable sets

If x0∊ An for any one n, then μx0 (An) = 2

⇒ ∑ ( )

= 1+0+0+ … = 1

Since x0∊ An, x0∊ ⋃

⇒ μx0 (⋃ ) = 1

Hence μx0 (⋃ ) = ∑

( )

Therefore μx0 is a measure and it is called a measure centered at x0.


Example 3.7.3

Let X be any non-empty set and fix a measurable set A0 ∊ X,

Define ( )

( )

Here is a measure which is called a measure concentrated on A0.

Definition 3.8.1. (Lebesque Integrable)

A complex valued measurable function f on X is said to be Lebesque integrable

If ∫

< ∞ and it is denoted by ( )

Definition 3.9.1. (Sub derivative of a function)

A sub derivative of a function f : I→R at a point in the open interval I is a real number

c such that ( ) ( ) ( )for all x in I.

Definition 3.9.2. (Sub derivative of a convex function)

A sub derivative of a convex function f : I→R at a point in the open interval I is a real

number c such that ( ) ( ) ( )for all x in I.

The set of sub derivatives at x0 for a convex function f is a nonempty closed interval

[a, b], where a and b are the one-sided limits

( ) ( )

( ) ( )

Which are guaranteed to exist and satisfy a ≤ b.


As in the picture, for any x0 in the domain of the function one can draw a line which goes

through the point (x0, f(x0)) and which is everywhere either touching or below the graph

of f. The slope of such a line is called a sub derivative (because the line is under the

graph of f).

Definition 3.9.3. (Sub differential of a function)

The set [a, b] of all sub derivatives is called the sub differential of the function f at x0.

Example 3.9.1

Consider the function f(x) = |x| which is convex. Then, the sub differential at the origin is

the interval [−1, 1].

The sub differential at any point x0 < 0 is the singleton set {−1},

∵ ( ) ( )

( )

And the sub differential at any point x0 > 0 is the singleton set {1}.

∵ ( ) ( )

( )

Note 3.9.1

If f is convex and its sub differential at contains exactly one sub derivative, then f is

differentiable at .


Note 3.9.2

Let f : I→R be a real-valued convex function defined on an open interval of the real line.

Such a function need not be differentiable at all points:

Example, the absolute value function f(x) = |x| is nondifferentiable when x = 0.but it is

convex.

Theorem 3.1.2. (Continuous-Jensen's Inequality).

Let μ be a positive measure on a σ-algebra Μ in a set Ω, so that μ(Ω) = 1. If f is a real

function in ( ) if a < f(x) < b for all x ϵ Ω , and if φ is convex on (a, b),

then (∫ ) ∫ ( )

Proof .

Given f ϵ ( ) (ie) f is a real-valued μ-integrable function on Ω,

Since φ is convex, at each real number x we have a nonempty set of sub derivatives,

which may be thought of as lines touching the graph of φ at x, but which are at or below

the graph of φ at all points.

By the definition of sub derivatives, there exist a real c such that

( ) ( ) ( )

⟹ ( ) ( )

⟹ ( ) ( ) -------- (1)

Let us choose and ( )

∴ (1) ⟹ ( ) for all real x and ( ) ------- (2)

Define ∫

-------- (3)

Now ( ) ( ( )) ( ) for all x


Taking integral over Ω ,

We have ∫

∫

∫

(Above is possible, because μ is a measure with μ(Ω)= 1 and f ϵ ( ) and by the

property of ( ), the integral is monotone)

∴ ∫

∫

∫

( ) ( by using (3))

( ) (∵ μ(Ω) =1)

∴ ∫

= ( ) ( by using (2) )

⟹ ∫

( )

⟹ ∫

(∫

) ( by using (3))

Definition 3.9.3 (Conjugate exponents)

If p and q are positive real numbers such that p + q = pq, or equivalently

, then

we call p and q a pair of conjugate exponents.

If both p & q ≤ 1, then

, hence either p or q >1

If both p & q = ∞, then

, hence either p or q < ∞

Example 3.9.3

If p = 1 then we should take q = ∞, then

,

Hence we can consider 1, ∞ as a pair of conjugate exponents.

Example 3.9.4

If p and q are integers, then the only conjugate exponents is 2, 2

Theorem 3.1.3 (Holder’s, Minkowski’s and Schwarz inequalities)


If p and q are conjugate exponents, 1 < p < ∞. Let X be a measure space with measure μ.

Let f and g are measurable functions on X with range in [0 ∞]. Then

∫

{ ∫

}

{∫

}

------------ (1)

{∫ ( )

}

{ ∫

}

{∫

}

-------- (2)

And ∫

{ ∫

}

{∫

}

-------------- (3)

(1) is called Holder’s inequality

(2) is called Minkowski’s inequality

(3) is called Schwarz’s inequality

Proof.

Let A = { ∫

}

and B = {∫

}

Case (i) Suppose A = 0, then { ∫

}

⟹ f = 0 almost every where

⟹ fg = 0 almost every where

∴ ∫

(ie) L.H.S of (1) = 0

R.H.S of (1) = AB = 0 (∵ A = 0)

Hence (1) holds for A = 0

Case (ii) Suppose A > 0 and B = ∞ ,

B = {∫

}

⟹ g = ∞

⟹ fg = ∞


∴ ∫

(ie) L.H.S of (1) = ∞

R.H.S of (1) = AB = ∞ (∵ B = ∞)

Hence (1) holds for A > 0 and B = ∞

Case (iii) Consider 0 < A < ∞, 0 < B < ∞

Let F =

and G =

Now ∫

∫ .

/

=

∫

6{ ∫ } 7

∫

∫

∫

∴ ∫

-------------(I)

Now ∫

∫ .

/

=

∫

6{ ∫ } 7

∫

∫

∫

∴ ∫

-------------- (II)

For each x ϵ X is such that 0 < F(x) < ∞ and, 0< G(x) < ∞ , there are real numbers s and t

such that ( ) ( )

⟹ ( )

( )

⟹ ( )

( )


Since p and q are conjugate exponents ,

, and by the convexity of ,

We have

⟹

(

)

(

)

⟹ ( ) ( )

* ( )+

* ( )+ for every x ϵ X

Taking integration,

∫

∫

∫

⟹ ∫

( by using (I) and (II) )

⟹ ∫

, (∵ p and q are conjugate exponents)

⟹ ∫ .

/ .

/

, (∵ F =

and G =

)

⟹

∫

⟹∫

⟹ ∫

{ ∫

}

{∫

}

, ( ∵ A = { ∫

}

, B ={∫

}

)

Hence (1) is holds for all A,B

To prove (2) :

Now ( ) ( ) ( )

Taking integration,

⟹ ∫ ( )

∫ ( )

∫ ( )

--------- (III)

Now ∫ ( )

{ ∫

}

{∫ ,( ) -

}

---------- (IV)


And ∫ ( )

{ ∫

}

{∫ ,( ) -

}

---------- (V)

(by using Holder’s inequality)

∴ Fom (III), (IV) and (V) ,

∫ ( )

{ ∫

}

{∫ ,( ) -

}

{ ∫

}

{∫ ,( ) -

}

{∫ ,( ) -

}

[{ ∫

}

{ ∫

}

]

{∫ ( )( )

}

[{ ∫

}

{ ∫

}

] --------- (VI)

But ( ) (∵ p +q = pq)

From (VI),

∫ ( )

{∫ ( )

}

[{ ∫

}

{ ∫

}

]

Divide both sides by {∫ ( )

}

,

∫ ( )

{∫ ( ) }

{ ∫

}

{ ∫

}

But

, which gives

Hence ∫ ( )

{∫ ( ) }

{ ∫

}

{ ∫

}

⟹{∫ ( )

}{∫ ( )

}

{∫ ( )

}

{ ∫

}

{ ∫

}

⟹ {∫ ( )

}

{ ∫

}

{ ∫

}

Hence (2) is proved


To prove (3) :

Let us take p = q = 2 in (1), we will get

∫

{ ∫

}

{∫

}

Hence (3) is proved.


4. Concave functions and its properties

Definition 4.10.1 (Concave function)

A real function φ defined on a segment (a, b), where -∞ ≤ a < b ≤ ∞ is called concave if

the inequality (( ) ) ( ) ( ) ( ) holds whenever a < x < b, a

< y < b, and 0< λ< 1.

Definition 4.10.2 (Concave function)


( ) ( ) ( )

Example. 4.10.1

The quadratic function ( ) for any real number x is concave.

Solution

To prove φ is concave, (i.e) to prove ( ( ) ) ( ) ( ) ( ) for

any λ such that and any real x,y


(i.e) to prove ( ( ) ) ( )

(i.e) to prove ( ( ) ) ( )

(i.e) to prove ( ( ) ) ( ( ) )

Now ( ( ) ) ( ( ) )

( ) ( ) ( )

( ) (( ) ( )) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )( )

( )( )

Clearly ( ) ≥ 0 for all real x,y

Since

∴

Hence ( ( ) ) ( ( ) ) ( )( )

∴ ( ) is a concave function

Definition 4.11.1 (Strictly concave function)

A real function φ defined on a segment (a, b), where -∞ ≤ a < b ≤ ∞ is called concave if

the inequality (( ) ) ( ) ( ) ( ) holds whenever a < x < b, a

< y < b, and 0< λ< 1.


Definition 4.11.2 (Strictly concave function)


( ) ( ) ( )

Lemma 4.1.15

Let φ be a concave function defined on some interval I of R, and let

, - , - be nondegenerate subintervals of I. That is,

and Assume that lies to the left of . That is, . Then

the slope of the chord over is greater than the slope of the chord over .. In particular,

( ) ( )

( ) ( )

( ) ( )

Proof.

Since

Let

Then

∴ ( )

∴φ( ) = φ( ( ) )

≥ ( ) ( ) ( ) (by the concavity of φ)

∴ φ( ) ≥ ( ) ( ) ( )

Subtract both sides by ( ),

φ( ) ( ) ≥ ( ) ( ) ( ) ( )

≥ ( ) ( ) ( ) ( )


≥( )( ( ) ( ))

≥

( ( ) ( )) , (∵ =

)

⟹ φ( ) ( ) ≥

( ( ) ( ))

⟹ ( ) ( )

≥ ( ) ( )

Similarly

Let

Then

∴ ( )

⟹ ( ) ( ( ) )

≥ ( ) ( ) ( ) (by the concavity of φ)

⟹ ( )≥ ( ) ( ) ( )

⟹ ( ) ( ) ( ) ( )

( )

( ) ( by using the value of λ and (1-λ) )

⟹ ( )

( )

( )

Adding both sides by ( ),

( ) ( )

( )

( ) ( )

( ) ( )

( ) .

/ ( )

( ) .

/ ( )

( )

( )


( )

( )

∴ ( ) ( )

( ) ( )

⟹ ( ) ( )

( ) ( )

Lemma 4.1.16

A real differentiable function φ is concave in (a, b) if and only if the second derivative is

negative (i.e) φ’’ ≤ 0.

Proof:

Using Taylor series

( ) ( ) ( )( ) ( )( )

(Approximately)

Suppose second derivative of φ is negative, (ie) ( )

Clearly ( )( )

Hence ( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( ) ---------- (1)

Put t = x in (1)

( ) ( ) ( )( ) ------- (2)


( ) ( ) ( )( ( ( ) ) )

( ) ( )( ( ) )

( ) ( )(( ) ( ) )

( ) ( ) ( )(( )( ) ) ---------- (3)

Put t = y in (1)


( ) ( ) ( )( ) ------- (4)


( ) ( ) ( )( ( ( ) ))

( ) ( )( ( ) )

( ) ( )( )

( ) ( ) ( )( ( )) ------------- (5)

Now ( ) ( ) ( )

⟹ ( ) ( ) ( )

{ ( ) ( )(( )( ))} ( ){ ( ) ( )( ( ))+

( ) ( )(( )( )) ( ) ( ) ( ) ( )( ( ))

( ) ( )( ( )( )) ( ) ( ) ( )( ( )( ))

( )

∴ ( ) ≥ ( ) ( ) ( )

Put ( )

Hence ( ( ) ) ≥ ( ) ( ) ( )

⟹ φ is concave


Example 4.10.2

The function φ( ) √ is concave for all real x

Solution

Given φ( ) √

⟹ ( )


⟹ ( ) .

/4

5

< 0 (Negative)

⟹ φ( ) √ is concave

Example 4.10.3

The logarithmic function ( ) is concave for all real x

Solution

Given ( )

⟹ ( )

⟹ ( )

< 0 (Negative)

⟹ ( ) is concave.

Note 4.12.1 (Affine function)

A real valued function f is said to be a affine function, then ( ) where a and

b are any two constants.

Example 4.12.1

The affine function ( ) is both convex and concave

Solution

Given f(x) =

⟹ ( )

⟹ ( )

⟹ we can say ( ) and ( )

Hence ( ) is both convex and concave.

Example 12.2


Example 4.12.2

The constant function ( ) is both convex and concave

Since ( )

Example 4.12.3

The function ( ) is convex on the set where x ≥ 0 and concave on the set

where x ≤ 0.

Solution

Given ( )

⟹ ( )

⟹ ( )

If x ≥ 0 , then ( ) ≥ 0 , hence it is convex

If x ≤ 0 , then ( ) ≤ 0 , hence it is concave

Lemma 4.1.17

A real valued function φ is convex on the interval I if and only if φ is concave

Proof.

Suppose φ is convex

Let x, y be real and 0 < λ < 1, then

( ( ) ) ( ) ( ) ( )

⇔ ( ( ) ) ( ) ( ) ( )

⇔ oncave

https://en.wikipedia.org/wiki/Concave_function


5. Some Special Inequalities

Note 5.12.1

If a function f is said to be linear, then ( ) ( ) ( ) for any x,y, where

a and b are constants

Lemma 5.1.18

Let φ and ψ be any two functions defined on the interval I

If φ is convex and ψ is linear then is convex

Proof.

Let x,y be any two real , 0 < λ < 1

Now ( ( ) ) ( ( ( ) ))

( ( ) ( ) ( )) (∵ ψ is linear)

( ( )) ( ) ( ( ) (by using convexity of φ )

( )) ( ) ( )

⟹ is convex.

Lemma 5.1.19

For any function f , the exponential function ( ) ( ) is convex

Proof.

Let x, y be any two real numbers and 0 < λ < 1

Now ( ( ) ) ( ) ( ) ( )

To prove ( ( ) ) ( ) ( ) ( )

To prove ( ) ( ) ( ) ( ) ( ) ( )

(ie) to prove ( ) ( ) ( ) ( ) ( ) ( )


Consider the function ( ) ( ) , for t > 0 ----- (1)

To find the minimum value of G(t) :

Now ( )

Put ( ) ⟹

⟹

⟹ ⟹ t = 1

Now ( ) ( )

Hence at t =1, ( ) ( ) > 0, (∵ 0 < λ < 1)

∴ G(t) has its minimum value at t = 1

⟹ G(t) > 0 , for all t > 0

Put t = ( ) ( ) in (1)

⟹ ( ( ) ( )) ( ) ( ( ) ( )) ( ( ) ( ))

Multiply by ( ), we will get

( ) ( ( ) ( )) ( )( ) ( ) ( ( ) ( )) ( )( ( ) ( ))

( )( ) ( ) ( ) ( ) ( ) ( ( ) ( )

( )( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

Since G(t) > 0 , for all t > 0,We have ( ( ) ( ))

Also ( ) ( ( ) ( ))

⟹ ( )( ) ( ) ( ) ( ) ( )

⟹ ( )( ) ( ) ( ) ( ) ( )


⟹ ( ) ( ) is convex

Lemma 5.1.20

If ( ) , then ( ) is convex.

Proof.

As per Lemma 1.19, for any function ( ) ( ) ( ) is convex.

Take ( )

Hence ( ) is convex

Note that ( ) is a concave function. ( Example 10.3)

Theorem 5.1.4

Prove that ∑ ∏

∑

∑ ∑

∑

Proof.

Let X = ∏

∑

⟹ X = , (∵ a = )

⟹ X = ∏

∑

= ∑

∑

(∵ log ab = log a + log b)

= ∑

∑

(∵ log ab = b log a)

Since ∑

∑

∑

∑

= 1, and by using the Lemma 1.20,

X = ∑

∑

∑

∑


∑

∑

∑

∑

∴ ∏

∑

∑

∑

⟹∑ ∏

∑

∑∑

∑

⟹∑ ∏

∑

∑ ∑

∑

Theorem 5.1.5 (Shannon’s inequality)

Given ∑ ∑

, then .

/ ∑ .

/

,

Proof.

In Theorem 1.4, Take

Theorem 1.4 ⟹ ∑ ∏

∑

∑ ∑

∑

⟹ ∏ (

)

∑

∑

(

)

∑

⟹∏ (

)

∑

∑

∑

⟹∏ (

)

( ∵ given ∑

∑

)

⟹

∏ (

)

Taking log on both sides, we will get


.

/ (∏ (

)

)

∑ .

/

(∵ log ab = log a + log b)

∑ .

/ .

/

(∵ log ab = b log a)

∑

.

/

Hence .

/ ∑

.

/

Theorem 5.1.6 (Renyi’s inequality)

Given ∑ ∑

,then for α > 0 ,α≠ 1

( ) ∑

(

)

Proof.

Applying Theorem 1.4 with


∑

∑ ∑

∑

⟹ ∑ ∏

∑

∑ ∑

∑

⟹ ∑

∑ (

)

But

∴ ∑

∑ ( )

⟹ ∑

∑ ( ( ) )

, (∵ )

Take

∑

∑

, we will get


∑ (

∑

)

(

∑

)

∑ (

∑

( )

∑

)

⟹ ∑

∑

∑

( )∑

∑

, ( ∵∑ ∑

)

⟹∑

⟹∑

⟹ ∑

------ (1)

Since λ < 1 , we have and hence

(negative)

Multiply both sides of (1) by

we will get,

∑

Without loss in generality, take α = λ and instead of j = 1…n takes i = 1…m,

we will get ∑

Subtract both sides by

, we will get

∑

⟹ ∑

∑

(∵ ∑

)

⟹ ∑

(

)

( )

Theorem 5.1.7 (Generalization of Holder’s inequality – Discrete)

∑ ∏

∏ (∑

)

∑

Proof.


∑


∑

∑ ∑

∑


⟹ ∑ ∏

∑

∑

∑ ∑

∑

∑

⟹ ∑ ∏ 4

∑

5

∑ ∑

∑

, ( ∵ ∑

)

⟹ ∑ ∏. /

.∑ /

∑∑

∑

⟹

∑ ∏ . /

∑ ∏ .∑ /

∑

∑

∑

⟹

∑ ∏ . /

∑ ∏ .∑ /

∑

⟹

∑ ∏ . /

∑ ∏ .∑ /

( ∵ ∑

)

⟹ ∑ ∏

∏ (∑

)

Theorem 5.1.8 (Arithmetic-Geometric-Mean inequality)

G.M ≤ A.M , (i.e) (∏ )

∑

Proof.


Since , we have ∑ ( )


∑

∑ ∑

∑

⟹ ∏ ( )

∑

⟹ (∏ )

∑


6. Conclusion

In this paper we generalize a very general case by using a suitable convex,

concave function of a real variable and studying its properties leads to some remarkable

inequalities.

Convex functions are particularly easy to minimize (for example, any minimum

of a convex function is a global minimum). For this reason, there is a very rich theory for

solving convex optimization problems that has many practical applications (for example,

circuit design, controller design, modeling, etc.),

The present exposition begins with the quantitative and combinatorial aspects

with topological vector spaces. The applications of convexity in other parts of

mathematics are concerned; separation and support theorems are of special importance.

They are widely used in functional analysis and have been used in game theory, in the

theory of summability, and even to prove certain coloring theorems of graph theory.

I am presently concentrating on convex minimization, maximization problems

and quasiconvex minimization and maximization. The convex maximization problem is

especially important for studying the existence of maxima. Consider the restriction of a

convex function to a compact convex set: Then, on that set, the function attains its

constrained maximum only on the boundary. These are very useful in the theory of

harmonic functions, potential theory, and partial differential equations.


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remarkable inequalities based on convex and concave functions

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