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On the (p,h)-convex function and some integral inequalities

Abstract

In this paper, we introduce a new class of (p,h)-convex functions which generalize P-functions and convex, h,p,s-convex, Godunova-Levin functions, and we give some properties of the functions. Moreover, we establish the corresponding Schur, Jensen, and Hadamard types of inequalities.

MSC:35K65, 35B33, 35B40.

1 Introduction

Let I and J be intervals in R. To motivate our work, let us recall the definitions of some special classes of functions.

Definition 1 [1]

A function f:IR is said to be a Godunova-Levin function or belongs to the class Q(I) if f is non-negative and

f ( α x + ( 1 α ) y ) f ( x ) α + f ( y ) 1 α

for all x,yI and α(0,1).

The class Q(I) was firstly described in [1] by Godunova and Levin. Some further properties of it are given in [2, 3]. It has been known that non-negative convex and monotone functions belong to this class of functions.

Definition 2 [4]

Let s(0,1) be a fixed real number. A function f:[0,)[0,) is said to be an s-convex function (in the second sense) or belongs to the class K s 2 , if

f ( α x + ( 1 α ) y ) α s f(x)+ ( 1 α ) s f(y)

for all x,yI and α[0,1].

An s-convex function was introduced by Breckner [4] and a number of properties and connections with s-convexity (in the first sense) were discussed in [5]. Of course, s-convexity means just convexity when s=1.

Definition 3 [2]

A function f:IR is said to be a P-function or belongs to the class P(I), if f is non-negative and

f ( α x + ( 1 α ) y ) f(x)+f(y)

for all x,yI and α[0,1].

For some results on the class P(I), see [6, 7].

Definition 4 [8]

Let I be a p-convex set. A function f:IR is said to be a p-convex function or belongs to the class PC(I), if

f ( [ α x p + ( 1 α ) y p ] 1 p ) αf(x)+(1α)f(y)

for all x,yI and α[0,1].

Remark 1 [8]

An interval I is said to be a p-convex set if [ α x p + ( 1 α ) y p ] 1 p I for all x,yI and α[0,1], where p=2k+1 or p= n m , n=2r+1, m=2t+1, and k,r,tN.

Definition 5 [9]

Let h:JR be a non-negative and non-zero function. We say that f:IR is an h-convex function or that f belongs to the class SX(I), if f is non-negative and

f ( α x + ( 1 α ) y ) h(α)f(x)+h(1α)f(y)

for all x,yI and α(0,1).

The h- and p-convex functions were introduced by Varšanec, Zhang and Wan, and a number of properties and Jensen’s inequalities of the functions were established (cf. [8]). As one can see, the definitions of the P-function, convex, h,p,s-convex, Godunova-Levin functions have similar forms. This observation leads us to generalize these varieties of convexity.

2 Definitions and basic results

In this section, we give new definitions and properties of the (p,h)-convex function. Throughout this paper, we assume that (0,1)J, f and h are real non-negative functions defined on I and J, respectively, and the set I is p-convex when fghx(p,h,I) or fghv(p,h,I). We first give a definition of the new class of convex functions.

Definition 6 Let h:JR be a non-negative and non-zero function. We say that f:IR is a (p,h)-convex function or that f belongs to the class ghx(h,p,I), if f is non-negative and

f ( [ α x p + ( 1 α ) y p ] 1 p ) h(α)f(x)+h(1α)f(y)
(2.1)

for all x,yI and α(0,1). Similarly, if the inequality sign in (2.1) is reversed, then f is said to be a (p,h)-concave function or belong to the class ghv(h,p,I).

Remark 2 It can be obviously seen that if h(α)=α, then all non-negative p-convex and p-concave functions belong to ghx(h,p,I) and ghv(h,p,I), respectively; if h(α)=α and p=1, then all non-negative convex functions belong to ghx(h,p,I); if h(α)= 1 α and p=1, then Q(I)=ghx(h,p,I); if h(α)= α s , s(0,1), and p=1, then K s 2 ghx(h,p,I); if h(α)=1 and p=1, then P(I)ghx(h,p,I), and if p=1, then SX(I)ghx(h,p,I).

Example 1 Let h k (α)= α k , where k1 and α>0. If f is a function defined as f(x)= x p , where p is an odd number and x0, we then have

f ( [ α x p + ( 1 α ) y p ] 1 p ) αf(x)+(1α)f(y) h k (α)f(x)+ h k (1α)f(y),

and hence, f belongs to ghx( h k ,p,I).

Next, we discuss some interesting properties of (p,h)-convex (concave) functions, which include linearity, product, composition properties, and an ordered property of h and p. In addition, we give some interesting properties of the (p,h)-convex function, when h is a super(sub)-multiplicative function.

Property 1 If f,gghx(h,p,I) and λ>0, then f+g,λfghx(h,p,I). Similarly, if f,gghv(h,p,I) and λ>0, then f+g,λfghv(h,p,I).

Proof The proof immediately follows from the definitions of the classes ghx(h,p,I) and ghv(h,p,I). □

Property 2 Let h 1 and h 2 be non-negative functions defined on an interval J with h 2 h 1 in (0,1). If fghx( h 2 ,p,I), then fghx( h 1 ,p,I). Similarly, if fghv( h 1 ,p,I), then fghv( h 2 ,p,I).

Proof If fghx( h 2 ,p,I), then for any x,yI and α(0,1) we have

f ( [ α x p + ( 1 α ) y p ] 1 p ) h 2 ( α ) f ( x ) + h 2 ( 1 α ) f ( y ) h 1 ( α ) f ( x ) + h 1 ( 1 α ) f ( y ) ,

and hence, fghx( h 1 ,p,I). □

Property 3 Let fghx(h, p 1 ,I).

  1. (a)

    For I(0,1], if f is monotone increasing (monotone decreasing), and p 2 p 1 >0 or p 2 p 1 <0, and ( p 1 p 2 >0 or p 1 p 2 <0), then fghx(h, p 2 ,I).

  2. (b)

    For I[1,), if f is monotone increasing (monotone decreasing), and p 1 p 2 >0 or p 1 p 2 <0, and ( p 2 p 1 >0 or p 2 p 1 <0), then fghx(h, p 2 ,I).

Let fghv(h, p 1 ,I).

  1. (c)

    For I(0,1], if f is monotone increasing (monotone decreasing), and p 1 p 2 >0 or p 1 p 2 <0, and ( p 2 p 1 >0 or p 2 p 1 <0), then fghv(h, p 2 ,I).

  2. (d)

    For I[1,), if f is monotone increasing (monotone decreasing), and p 2 p 1 >0 or p 2 p 1 <0, and ( p 1 p 2 >0 or p 1 p 2 <0), then fghv(h, p 2 ,I).

Proof (a) Setting g(p)= ( α x p + ( 1 α ) y p ) 1 p , we have

g (p)= 1 p ( α x p + ( 1 α ) y p ) 1 p 1 ( α x p ln ( x ) + ( 1 α ) y p ln ( y ) ) .

When p>0 and x,y(0,1], we have g (p)<0, and so g( p 2 )g( p 1 ). We then obtain

f ( g ( p 2 ) ) f ( g ( p 1 ) ) h(α)f(x)+(1α)f(y),

since f is monotone increasing and fghx(h, p 1 ,I). Therefore, we get fghx(h, p 2 ,I).

The results of (b), (c), and (d) follow by similar arguments as above. □

Property 4 Let f and g be similarly ordered functions on I, i.e.,

( f ( x ) f ( y ) ) ( g ( x ) g ( y ) ) 0
(2.2)

for all x,yI. If fghx( h 1 ,p,I), gghx( h 2 ,p,I), and h(α)+h(1α)c for all α(0,1), where h(t)=max( h 1 (t), h 2 (t)) and c is a fixed positive number, then the product fg belongs to ghx(ch,p,I). Similarly, let f and g be oppositely ordered, i.e.,

( f ( x ) f ( y ) ) ( g ( x ) g ( y ) ) 0

for all x,yI. If fghv( h 1 ,p,I), gghv( h 2 ,p,I), and h(α)+h(1α)c for all α(0,1), where h(t)=min( h 1 (t), h 2 (t)) and c is a fixed positive number, then the product fg belongs to ghv(ch,p,I).

Proof We only give a proof for the first part, since the result of the second part of this theorem follows by a similar argument. By (2.2), we have

f(x)g(x)+f(y)g(y)f(x)g(y)+f(y)g(x).

Let α and β be positive numbers such that α+β=1. We then obtain

f g ( [ α x p + β y p ] 1 p ) ( h 1 ( α ) f ( x ) + h 1 ( β ) f ( y ) ) ( h 2 ( α ) g ( x ) + h 2 ( β ) g ( y ) ) h 2 ( α ) f g ( x ) + h ( α ) h ( β ) f ( x ) g ( y ) + h ( α ) h ( β ) f ( y ) g ( x ) + h 2 ( β ) f g ( y ) h 2 ( α ) f g ( x ) + h ( α ) h ( β ) f ( x ) g ( x ) + h ( α ) h ( β ) f ( y ) g ( y ) + h 2 ( β ) f g ( y ) = ( h ( α ) + h ( β ) ) ( h ( α ) f g ( x ) + h ( β ) f g ( y ) ) c h ( α ) f g ( x ) + c h ( β ) f g ( y ) ,

which completes the proof. □

Definition 7 [9]

A function h:IR is called a super-multiplicative function if

h(xy)h(x)h(y)
(2.3)

for all x,yJ.

If the inequality sign in (2.3) is reversed, then h is said to be a sub-multiplicative function, and if the equality holds in (2.3), then h is called a multiplicative function.

Example 2 Let h(x)=c e x . If c=1, then h is a multiplicative function. If c>1, then h is a sub-multiplicative function, and if 0<c<1, then h is a super-multiplicative function.

Property 5 Let I be an interval such that 0I. We then have the following.

  1. (a)

    If fghx(h,p,I), f(0)=0, and h is super-multiplicative, then the inequality

    f ( [ α x p + β y p ] 1 p ) h(α)f(x)+h(β)f(y)
    (2.4)

    holds for all x,yI and all α,β>0 such that α+β1.

  2. (b)

    Let h be a non-negative function with h(α)< 1 2 for some α(0, 1 2 ). If f is a non-negative function satisfying (2.4) for all x,yI and all α,β>0 with α+β1, then f(0)=0.

  3. (c)

    If fghv(h,p,I), f(0)=0, and h is sub-multiplicative, then the inequality

    f ( [ α x p + β y p ] 1 p ) h(α)f(x)+h(β)f(y)
    (2.5)

    holds for all x,yI and all α,β>0 such that α+β1.

  4. (d)

    Let h be a non-negative function with h(α)> 1 2 for some α(0, 1 2 ). If f is a non-negative function satisfying (2.5) for all x,yI and all α,β>0 with α+β1, then f(0)=0.

Proof (a) Let α,β>0, α+β=γ<1, and let a and b be numbers such that a= α γ and b= β γ . We then have a+b=1 and

f ( [ α x p + β y p ] 1 p ) = f ( [ a γ x p + b γ y p ] 1 p ) h ( a ) f ( γ 1 p x ) + h ( b ) f ( γ 1 p y ) = h ( a ) f ( [ γ x p + ( 1 γ ) 0 p ] 1 p ) + h ( b ) f ( [ γ y p + ( 1 γ ) 0 p ] 1 p ) h ( a ) h ( γ ) f ( x ) + h ( a ) h ( 1 γ ) f ( 0 ) + h ( b ) h ( γ ) f ( y ) + h ( b ) h ( 1 γ ) f ( 0 ) = h ( a ) h ( γ ) f ( x ) + h ( b ) h ( γ ) f ( y ) h ( a γ ) f ( x ) + h ( b γ ) f ( y ) = h ( α ) f ( x ) + h ( β ) f ( y ) .

(b) If f(0)0, then f(0)>0. Setting x=y=0 in (2.4), we get

f(0)h(α)f(0)+h(β)f(0).

By setting α=β, where α(0, 1 2 ), and dividing both sides of the inequality above by f(0), we obtain 2h(α)1 for all α(0, 1 2 ), which is a contradiction to the assumption h(α)< 1 2 for some α(0, 1 2 ), and so f(0)=0.

The results of (c) and (d) follow by using similar arguments as above, and so we omit the proofs here. □

Corollary 1 Let h s (x)= x s , where s,x>0, and let 0I. For all fghx( h s ,p,I), inequality (2.4) holds for all α,β>0 with α+β1 if and only if f(0)=0. For all fghv( h s ,p,I), inequality (2.5) holds for all α,β>0 with α+β1 if and only if f(0)=0.

Proof Let α,β>0, α+β=γ<1, and let a and b be positive numbers such that a= α γ and b= β γ . We then have a+b=1 and

f ( [ α x p + β y p ] 1 p ) = f ( [ a γ x p + b γ y p ] 1 p ) a s f ( γ 1 p x ) + b s f ( γ 1 p y ) = a s f ( [ γ x p + ( 1 γ ) 0 p ] 1 p ) + b s f ( [ γ y p + ( 1 γ ) 0 p ] 1 p ) a s γ s f ( x ) + a s ( 1 γ ) s f ( 0 ) + b s γ s f ( y ) + b s ( 1 γ ) s f ( 0 ) = a s γ s f ( x ) + b s γ s f ( y ) = α s f ( x ) + β s f ( y ) .

Setting x=y=α=β=0 in (2.4), we get f(0)0, while f(0)0 by the definition of the (p,h)-convex function, and hence f(0)=0. □

Property 6 Suppose that h i : J i (0,), i=1,2, are functions such that h 2 ( J 2 ) J 1 and h 2 (α)+ h 2 (1α)1 for all α(0,1), and that f: I 1 [0,) and g: I 2 [0,) are functions with g( I 2 ) I 1 , 0 I 1 , and f(0)=0.

If h 1 is a super-multiplicative function, fSX( h 1 , I 1 ), and f is increasing (decreasing) and gghx( h 2 ,p, I 2 ) (gghv( h 2 ,p, I 2 )), then the composite function fg belongs to ghx( h 1 h 2 ,p, I 2 ). If h 1 is a sub-multiplicative function, fSV( h 1 , I 1 ), and f is increasing (decreasing) and gghv( h 2 ,p, I 2 ) (gghx( h 2 ,p, I 2 )), then the composite function fg belongs to ghv( h 1 h 2 ,p, I 2 ).

Proof If gghx( h 2 ,p, I 2 ) and f is an increasing function, then we have

(fg) ( [ α x p + ( 1 α ) y p ] 1 p ) f ( h 2 ( α ) g ( x ) + h 2 ( 1 α ) g ( y ) )

for all x,y I 2 and α(0,1). Using Property 5(a) with p=1, we obtain

f ( h 2 ( α ) g ( x ) + h 2 ( 1 α ) g ( y ) ) h 1 ( h 2 ( α ) ) f ( g ( x ) ) + h 1 ( h 2 ( 1 α ) ) f ( g ( y ) ) ,

which implies that fg belongs to ghx( h 1 h 2 ,p, I 2 ). □

If f is a convex or concave function, then we may give a similar statement on the composite function of f and g.

Property 7 Let f: I 1 [0,) and g: I 2 [0,) be functions with g( I 2 ) I 1 . If the function f is convex and increasing (decreasing), and gghx(h,p, I 2 ) (gghv(h,p, I 2 )) with h(α)+h(1α)=1 for α(0,1), then fg belongs to ghx(h,p, I 2 ). If the function f is concave and increasing (decreasing), and gghv(h,p, I 2 ) (gghx(h,p, I 2 )) with h(α)+h(1α)=1 for α(0,1), then fg belongs to ghv(h,p, I 2 ).

Proof If gghx(h,p, I 2 ) and f is an increasing function, we then have

(fg) ( [ α x p + ( 1 α ) y p ] 1 p ) f ( h ( α ) g ( x ) + h ( 1 α ) g ( y ) )

for all x,y I 2 and α(0,1). Since h(α)+h(1α)=1 and f is convex, we obtain

f ( h ( α ) g ( x ) + h ( 1 α ) g ( y ) ) h(α)f ( g ( x ) ) +h(1α)f ( g ( y ) ) ,

which implies that fg belongs to ghx(h,p, I 2 ). □

3 Schur-type inequalities

In this section, we establish Schur-type inequalities of (p,h)-convex functions.

Theorem 1 Let h:JR be a non-negative super-multiplicative function and let f:IR be a function such that fghx(h,p,I). Then for all x 1 , x 2 , x 3 I such that x 1 < x 2 < x 3 and x 3 p x 1 p , x 3 p x 2 p , x 2 p x 1 p J, the following inequality holds:

h ( x 3 p x 2 p ) f( x 1 )h ( x 3 p x 1 p ) f( x 2 )+h ( x 2 p x 1 p ) f( x 3 )0.
(3.1)

If the function h is sub-multiplicative and fghv(h,p,I), then the inequality sign in (3.1) is reversed.

Proof Let fghx(h,p,I) and let x 1 , x 2 , x 3 I be the numbers stated in this theorem. Then one can easily see that

x 3 p x 2 p x 3 p x 1 p , x 2 p x 1 p x 3 p x 1 p (0,1)Jand x 3 p x 2 p x 3 p x 1 p + x 2 p x 1 p x 3 p x 1 p =1.

We also have

h ( x 3 p x 2 p ) =h ( x 3 p x 2 p x 3 p x 1 p ( x 3 p x 1 p ) ) h ( x 3 p x 2 p x 3 p x 1 p ) h ( x 3 p x 1 p )

and

h ( x 2 p x 1 p ) h ( x 2 p x 1 p x 3 p x 1 p ) h ( x 3 p x 1 p ) .

Setting α= x 3 p x 2 p x 3 p x 1 p , x= x 1 , and y= x 3 in (2.1), we have x 2 p =α x p +(1α) y p and

f ( x 2 ) h ( x 3 p x 2 p x 3 p x 1 p ) f ( x 1 ) + h ( x 2 p x 1 p x 3 p x 1 p ) f ( x 3 ) h ( x 3 p x 2 p ) h ( x 3 p x 1 p ) f ( x 1 ) + h ( x 2 p x 1 p ) h ( x 3 p x 1 p ) f ( x 3 ) .
(3.2)

Assuming h( x 3 p x 1 p )>0 and multiplying both sides of the inequality above by h( x 3 p x 1 p ), we obtain inequality (3.1). □

Remark 3 In fact, if f(x)= x λ , λR, h(x)= h 1 (x)= 1 x , p=1, and x 1 , x 2 , x 3 I=(0,1), then inequality (3.1) gives the Schur inequality, see [[10], p.177].

The following corollary gives a Schur-type inequality for the (p,h)-convex function.

Corollary 2 If f:I=(0,1)I belongs to the class ghx( h k ,p,I) and h k = 1 x k , then we have the inequality

f ( x 1 ) ( x 3 p x 1 p ) k ( x 2 p x 1 p ) k f ( x 2 ) ( x 3 p x 2 p ) k ( x 2 p x 1 p ) k + f ( x 3 ) ( x 3 p x 1 p ) k ( x 3 p x 2 p ) k 0
(3.3)

for all x 1 , x 2 , x 3 I with x 1 < x 2 < x 3 . If fghv( h k ,p,I), then the inequality sign in (3.3) is reversed. If k=1, p=1, and f(x)= x λ , λR, then fghx( h 1 ,1,I) and inequality (3.3) gives the Schur inequality.

4 Jensen-type inequalities

In this section, we introduce some Jensen-type inequalities of (p,h)-convex functions.

Theorem 2 Let w 1 ,, w n be positive real numbers with n2. If h is a non-negative super-multiplicative function and if fghx(h,p,I) and x 1 ,, x n I, then we have the inequality

f ( [ 1 W n i = 1 n w i x i p ] 1 p ) i = 1 n h ( w i W n ) f( x i ),where  W n = i = 1 n w i .
(4.1)

If h is sub-multiplicative and fghv(h,p,I), then the inequality sign in (4.1) is reversed.

Proof When n=2, inequality (4.1) holds by (2.1) with α= w 1 W 2 . Assuming inequality (4.1) holds for n1, we obtain

f ( [ 1 W n i = 1 n w i x i p ] 1 p ) = f ( [ w n W n x n p + i = 1 n 1 w i W n x i p ] 1 p ) = f ( [ w n W n x n p + W n 1 W n i = 1 n 1 w i W n 1 x i p ] 1 p ) h ( w n W n ) f ( x n ) + h ( W n 1 W n ) f ( [ i = 1 n 1 w i W n 1 x i p ] 1 p ) h ( w n W n ) f ( x n ) + h ( W n 1 W n ) i = 1 n 1 h ( w i W n 1 ) f ( x i ) i = 1 n h ( w i W n ) f ( x i ) ,

and, hence, the result follows by mathematical induction. □

Remark 4 For h(α)=α and p=1, inequality (4.1) becomes the classical Jensen inequality.

Theorem 3 Let w 1 ,, w n be positive real numbers and let (m,M) be an interval in I. If h:(0,)R is a non-negative super-multiplicative function and fghx(h,p,I), then for all x 1 ,, x n (m,M) we have the inequality

i = 1 n h ( w i W n ) f ( x i ) f ( m ) i = 1 n h ( w i W n ) h ( M p x i p M p m p ) + f ( M ) i = 1 n h ( w i W n ) h ( x i p m p M p m p ) .
(4.2)

If h is a non-negative sub-multiplicative function and fghv(h,p,I), then the inequality sign in (4.2) is reversed.

Proof Setting x 1 =m, x 2 = x i , and x 3 =M in (3.2), we get the inequalities

f( x i )h ( M p x i p M p m p ) f(m)+h ( x i p m p M p m p ) f(M),i=1,,n.

Multiplying both sides of the above inequality with h( w i W n ) and adding all inequalities side by side for i=1,,n, we obtain (4.2). □

Let K be a finite nonempty set of positive integers and let F be an index set function defined by

F(K)=h( W K )f ( [ 1 W K i K w i x i p ] 1 p ) i K h( w i )f( x i ),where  W K = i K w i .

Theorem 4 Let h:(0,)R be a non-negative function, and let M and K be finite nonempty sets of positive integers such that MK=. If h is super-multiplicative and f:IR belongs to the class ghx(h,p,I), then for w i >0, x i I, iMK we have the inequality

F(MK)F(M)+F(K).
(4.3)

If h is sub-multiplicative and fghv(h,p,I), then the inequality sign in (4.3) is reversed.

Proof Setting x= [ 1 W M i M w i x i p ] 1 p , y= [ 1 W K i K w i x i p ] 1 p , and α= W M W M K in (2.1), we obtain the inequality

f ( [ 1 W M K i M K w i x i p ] 1 p ) h ( W M W M K ) f ( [ 1 W M i M w i x i p ] 1 p ) + h ( W K W M K ) f ( [ 1 W K i K w i x i p ] 1 p ) .

Multiplying both sides of the above inequality with h( W M K ), we get the inequality

h ( W M K ) f ( [ 1 W M K i M K w i x i p ] 1 p ) h ( W M ) f ( [ 1 W M i M w i x i p ] 1 p ) + h ( W K ) f ( [ 1 W K i K w i x i p ] 1 p ) .

Subtracting i M K h( w i )f( x i ) from both sides of the inequality above and using the identity i M K h( w i )f( x i )= i M h( w i )f( x i )+ i K h( w i )f( x i ), we obtain (4.3). □

A simple consequence of Theorem 4 is stated in the following corollary without proof.

Corollary 3 Let h:(0,)R be a non-negative super-multiplicative function. If w i >0, i=1,,n, and M k ={1,,K}, then for fghx(h,p,I) we have

F( M n )F( M n 1 )F( M 2 )0
(4.4)

and

F( M n ) min 1 i < j n { h ( w i + w j ) f ( [ w i x i p + w j x j p w i + w j ] 1 p ) h ( w i ) f ( x i ) h ( w j ) f ( x j ) } .
(4.5)

If h is sub-multiplicative and fghv(h,p,I), then the inequality signs in (4.4) and (4.5) are reversed, and min is replaced with max.

Remark 5 Some results obtained from Theorem 4 and Corollary 3 are given in [[11], p.7], when h(α)=α, p=1, and h is a convex or concave function.

5 Hadamard-type inequalities

In this section, we give some Hadamard-type inequalities of (p,h)-convex functions.

Theorem 5 If fghx(h,p,I) L 1 ([a,b]) for a,bI with a<b, then we have

1 2 h ( 1 2 ) f ( [ a p + b p 2 ] 1 p ) p b p a p a b x p 1 f(x)dx ( f ( a ) + f ( b ) ) 0 1 h(t)dt.
(5.1)

Proof Setting x p = y a b a b p + b y b a a p , we get

p b p a p a b x p 1 f(x)dx= 1 b a a b f ( [ y a b a b p + b y b a a p ] 1 p ) dy.

By using inequality (2.1) we obtain

f ( [ y a b a b p + b y b a a p ] 1 p ) h ( y a b a ) f(b)+h ( b y b a ) f(a),

and hence, by integrating the above inequality over [a,b], we have

a b f ( [ y a b a b p + b y b a a p ] 1 p ) d y f ( b ) a b h ( y a b a ) d y + f ( a ) a b h ( b y b a ) d y = ( b a ) ( f ( a ) + f ( b ) ) 0 1 h ( t ) d t ,

which gives the second inequality.

Setting y= 1 2 (a+b)+t, we obtain

1 2 ( b a ) 1 2 ( b a ) f ( [ 1 2 ( a p + b p ) + b p a p b a t ] 1 p ) d t = 0 1 2 ( b a ) f ( [ 1 2 ( a p + b p ) + b p a p b a t ] 1 p ) d t + 0 1 2 ( b a ) f ( [ 1 2 ( a p + b p ) b p a p b a t ] 1 p ) d t 1 h ( 1 / 2 ) 0 1 2 ( b a ) f ( [ 1 2 ( a p + b p ) ] 1 p ) d t = b a 2 h ( 1 / 2 ) f ( [ 1 2 ( a p + b p ) ] 1 p ) ,

and, hence, the first inequality follows. □

Remark 6 If h(α)=α and p=1, then inequality (5.1) gives the classical Hadamard inequality.

Theorem 6 Suppose that f and g are functions such that fghx( h 1 ,p,I), gghx( h 2 ,p,I), fg L 1 ([a,b]), and h 1 h 2 L 1 ([0,1]) with a,bI and a<b. We then have

p b p a p a b x p 1 f ( x ) g ( x ) d x M ( a , b ) 0 1 h 1 ( t ) h 2 ( t ) d t + N ( a , b ) 0 1 h 1 ( t ) h 2 ( 1 t ) d t ,
(5.2)

where M(a,b)=f(a)g(a)+f(b)g(b) and N(a,b)=f(a)g(b)+f(b)g(a).

Proof Since fghx( h 1 ,p,I) and gghx( h 2 ,p,I), we have

f ( [ t a p + ( 1 t ) b p ] 1 p ) h 1 ( t ) f ( a ) + h 1 ( 1 t ) f ( b ) , g ( [ t a p + ( 1 t ) b p ] 1 p ) h 2 ( t ) g ( a ) + h 2 ( 1 t ) g ( b )

for all t[0,1]. Because f and g are non-negative, we get the inequality

f ( [ t a p + ( 1 t ) b p ] 1 p ) g ( [ t a p + ( 1 t ) b p ] 1 p ) h 1 ( t ) h 2 ( t ) f ( a ) g ( a ) + h 1 ( 1 t ) h 2 ( t ) f ( b ) g ( a ) + h 1 ( t ) h 2 ( 1 t ) f ( a ) g ( b ) + h 1 ( 1 t ) h 2 ( 1 t ) f ( b ) g ( b ) .

Integrating both sides of the above inequality over (0,1), we obtain the inequality

0 1 f ( [ t a p + ( 1 t ) b p ] 1 p ) g ( [ t a p + ( 1 t ) b p ] 1 p ) d t f ( a ) g ( a ) 0 1 h 1 ( t ) h 2 ( t ) d t + f ( b ) g ( a ) 0 1 h 1 ( 1 t ) h 2 ( t ) d t + f ( a ) g ( b ) 0 1 h 1 ( t ) h 2 ( 1 t ) d t + f ( b ) g ( b ) 0 1 h 1 ( 1 t ) h 2 ( 1 t ) d t .

Setting x= [ t a p + ( 1 t ) b p ] 1 p , we get

p b p a p a b x p 1 f(x)g(x)dxM(a,b) 0 1 h 1 (t) h 2 (t)dt+N(a,b) 0 1 h 1 (t) h 2 (1t)dt.

 □

Theorem 7 Let fghx( h 1 ,p,I), gghx( h 2 ,p,I) be functions such that fg L 1 ([a,b]) and h 1 h 2 L 1 ([0,1]), and let a,bI with a<b. We then have

1 2 h 1 ( 1 2 ) h 2 ( 1 2 ) f ( [ a p + b p 2 ] 1 p ) g ( [ a p + b p 2 ] 1 p ) p b p a p a b x p 1 f ( x ) g ( x ) d x M ( a , b ) 0 1 h 1 ( t ) h 2 ( 1 t ) d t + N ( a , b ) 0 1 h 1 ( t ) h 2 ( t ) d t .
(5.3)

Proof Since a p + b p 2 = t a p + ( 1 t ) b p 2 + ( 1 t ) a p + t b p 2 , we have

f ( [ a p + b p 2 ] 1 p ) g ( [ a p + b p 2 ] 1 p ) = f ( [ t a p + ( 1 t ) b p 2 + ( 1 t ) a p + t b p 2 ] 1 p ) g ( [ t a p + ( 1 t ) b p 2 + ( 1 t ) a p + t b p 2 ] 1 p ) = h 1 ( 1 2 ) [ f ( [ t a p + ( 1 t ) b p ] 1 p ) + f ( [ ( 1 t ) a p + t b p ] 1 p ) ] × h 2 ( 1 2 ) [ g ( [ t a p + ( 1 t ) b p ] 1 p ) + g ( [ ( 1 t ) a p + t b p ] 1 p ) ] h 1 ( 1 2 ) h 2 ( 1 2 ) [ f ( [ t a p + ( 1 t ) b p ] 1 p ) g ( [ t a p + ( 1 t ) b p ] 1 p ) ] + h 1 ( 1 2 ) h 2 ( 1 2 ) [ f ( [ ( 1 t ) a p + t b p ] 1 p ) g ( [ ( 1 t ) a p + t b p ] 1 p ) ] + h 1 ( 1 2 ) h 2 ( 1 2 ) [ h 1 ( t ) f ( a ) + h 1 ( 1 t ) f ( b ) ] [ h 2 ( 1 t ) g ( a ) + h 2 ( t ) g ( b ) ] + h 1 ( 1 2 ) h 2 ( 1 2 ) [ h 1 ( 1 t ) f ( a ) + h 1 ( t ) f ( b ) ] [ h 2 ( t ) g ( a ) + h 2 ( 1 t ) g ( b ) ] = h 1 ( 1 2 ) h 2 ( 1 2 ) [ f ( [ t a p + ( 1 t ) b p ] 1 p ) g ( [ t a p + ( 1 t ) b p ] 1 p ) ] + h 1 ( 1 2 ) h 2 ( 1 2 ) [ f ( [ ( 1 t ) a p + t b p ] 1 p ) g ( [ ( 1 t ) a p + t b p ] 1 p ) ] + h 1 ( 1 2 ) h 2 ( 1 2 ) [ ( h 1 ( t ) h 2 ( 1 t ) + h 1 ( 1 t ) h 2 ( t ) ) M ( a , b ) ] + h 1 ( 1 2 ) h 2 ( 1 2 ) [ ( h 1 ( t ) h 2 ( t ) + h 1 ( 1 t ) h 2 ( 1 t ) ) N ( a , b ) ] .

Integrating the above inequality over [0,1], we obtain

1 2 h 1 ( 1 2 ) h 2 ( 1 2 ) f ( [ a p + b p 2 ] 1 p ) g ( [ a p + b p 2 ] 1 p ) p b p a p a b x p 1 f ( x ) g ( x ) d x M ( a , b ) 0 1 h 1 ( t ) h 2 ( 1 t ) d t + N ( a , b ) 0 1 h 1 ( t ) h 2 ( t ) d t .

 □

Theorem 8 Let fghx( h 1 ,p,I) and gghx( h 2 ,p,I) be functions such that fg L 1 ([a,b]), h 1 h 2 L 1 ([0,1]), and let a,bI with a<b. We then have the inequality

p 2 2 ( b p a p ) 2 a b a b 0 1 x p 1 y p 1 f ( [ t x p + ( 1 t ) y p ] 1 p ) g ( [ t x p + ( 1 t ) y p ] 1 p ) d x d y d t p b p a p 0 1 h 1 ( t ) h 2 ( t ) d t a b x p 1 f ( x ) g ( x ) d x + 0 1 h 1 ( t ) d t 0 1 h 2 ( t ) d t 0 1 h 1 ( t ) h 2 ( 1 t ) d t [ M ( a , b ) + N ( a , b ) ] .
(5.4)

Proof Since fghx( h 1 ,p,I) and gghx( h 2 ,p,I), we have

f ( [ t x p + ( 1 t ) y p ] 1 p ) h 1 ( t ) f ( x ) + h 1 ( 1 t ) f ( y ) , g ( [ t x p + ( 1 t ) y p ] 1 p ) h 2 ( t ) g ( x ) + h 2 ( 1 t ) g ( y )

for all t[0,1]. Because f and g are non-negative, we get the inequality

f ( [ t x p + ( 1 t ) y p ] 1 p ) g ( [ t x p + ( 1 t ) y p ] 1 p ) h 1 ( t ) h 2 ( t ) f ( x ) g ( x ) + h 1 ( 1 t ) h 2 ( t ) f ( y ) g ( x ) + h 1 ( t ) h 2 ( 1 t ) f ( x ) g ( y ) + h 1 ( 1 t ) h 2 ( 1 t ) f ( y ) g ( y ) .

Multiplying both sides of the above inequality with p 2 x p 1 y p 1 ( b p a p ) 2 and integrating the result over [a,b] and [0,1], we obtain the inequality

p 2 ( b p a p ) 2 a b a b 0 1 x p 1 y p 1 f ( [ t x p + ( 1 t ) y p ] 1 p ) g ( [ t x p + ( 1 t ) y p ] 1 p ) d x d y d t 0 1 h 1 ( t ) h 2 ( t ) d t [ p 2 ( b p a p ) 2 ( a b x p 1 f ( x ) g ( x ) d x a b y p 1 d y + a b y p 1 f ( y ) g ( y ) d y a b x p 1 d x ) ] + 2 0 1 h 1 ( t ) h 2 ( 1 t ) d t [ p 2 ( b p a p ) 2 a b x p 1 f ( x ) d x a b y p 1 f ( y ) d y ] .

By (5.1), we have the inequality

p 2 2 ( b p a p ) 2 a b a b 0 1 x p 1 y p 1 f ( [ t x p + ( 1 t ) y p ] 1 p ) g ( [ t x p + ( 1 t ) y p ] 1 p ) d x d y d t p b p a p 0 1 h 1 ( t ) h 2 ( t ) d t a b x p 1 f ( x ) g ( x ) d x + 0 1 h 1 ( t ) d t 0 1 h 2 ( t ) d t 0 1 h 1 ( t ) h 2 ( 1 t ) d t [ M ( a , b ) + N ( a , b ) ] .

 □

Theorem 9 Let fghx( h 1 ,p,I), gghx( h 2 ,p,I) be functions such that fg L 1 ([a,b]), h 1 h 2 L 1 ([0,1]), and let a,bI with a<b. We then have the inequality

a b 0 1 x p 1 f ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) g ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) d t d x 0 1 h 1 ( t ) h 2 ( t ) d t a b x p 1 f ( x ) g ( x ) d x + b p a p p [ M ( a , b ) + N ( a , b ) ] × [ h 1 ( 1 2 ) h 2 ( 1 2 ) 0 1 h 1 ( t ) h 2 ( t ) d t + [ h 1 ( 1 2 ) 0 1 h 2 ( t ) d t + h 2 ( 1 2 ) 0 1 h 1 ( t ) d t ] 0 1 h 1 ( t ) h 2 ( 1 t ) d t ] .
(5.5)

Proof Since fghx( h 1 ,p,I) and gghx( h 2 ,p,I), we have the inequalities

f ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) h 1 ( t ) f ( x ) + h 1 ( 1 t ) f ( [ a p + b p 2 ] 1 p ) , g ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) h 2 ( t ) g ( x ) + h 2 ( 1 t ) g ( [ a p + b p 2 ] 1 p )

for all t[0,1]. Because f and g are non-negative, we get the inequality

f ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) g ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) h 1 ( t ) h 2 ( t ) f ( x ) g ( x ) + h 1 ( 1 t ) h 2 ( t ) f ( [ a p + b p 2 ] 1 p ) g ( x ) + h 1 ( t ) h 2 ( 1 t ) f ( x ) g ( [ a p + b p 2 ] 1 p ) + h 1 ( 1 t ) h 2 ( 1 t ) f ( [ a p + b p 2 ] 1 p ) g ( [ a p + b p 2 ] 1 p ) .

Multiplying both sides of the inequality above with x p 1 and integrating the result over [a,b] and [0,1], we obtain

a b 0 1 x p 1 f ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) g ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) d t d x 0 1 h 1 ( t ) h 2 ( t ) d t [ a b x p 1 f ( x ) g ( x ) d x + b p a p p f ( [ a p + b p 2 ] 1 p ) g ( [ a p + b p 2 ] 1 p ) ] + 0 1 h 1 ( t ) h 2 ( 1 t ) d t [ g ( [ a p + b p 2 ] 1 p ) a b x p 1 f ( x ) d x + f ( [ a p + b p 2 ] 1 p ) a b x p 1 g ( x ) d x ] .

By inequality (5.1), we have

a b 0 1 x p 1 f ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) g ( [ t x p + ( 1 t ) a p + b p 2 ] 1 p ) d t d x 0 1 h 1 ( t ) h 2 ( t ) d t [ a b x p 1 f ( x ) g ( x ) d x + b p a p p h 1 ( 1 2 ) ( f ( a ) + f ( b ) ) h 2 ( 1 2 ) ( g ( a ) + g ( b ) ) ] + 0 1 h 1 ( t ) h 2 ( 1 t ) d t b p a p p h 2 ( 1 2 ) ( f ( a ) + f ( b ) ) ( g ( a ) + g ( b ) ) 0 1 h 1 ( t ) d t + 0 1 h 1 ( t ) h 2 ( 1 t ) d t b p a p p h 1 ( 1 2 ) ( f ( a ) + f ( b ) ) ( g ( a ) + g ( b ) ) 0 1 h 2 ( t ) d t = 0 1 h 1 ( t ) h 2 ( t ) d t a b x p 1 f ( x ) g ( x ) d x + b p a p p [ M ( a , b ) + N ( a , b ) ] × [ h 1 ( 1 2 ) h 2 ( 1 2 ) 0 1 h 1 ( t ) h 2 ( t ) d t + [ h 1 ( 1 2 ) 0 1 h 2 ( t ) d t + h 2 ( 1 2 ) 0 1 h 1 ( t ) d t ] 0 1 h 1 ( t ) h 2 ( 1 t ) d t ] .

 □

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Acknowledgements

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

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Fang, Z.B., Shi, R. On the (p,h)-convex function and some integral inequalities. J Inequal Appl 2014, 45 (2014). https://doi.org/10.1186/1029-242X-2014-45

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Keywords

  • (p,h)-convex function
  • Schur-type inequality
  • Jensen-type inequality
  • Hadamard-type inequality