# Hermite-Hadamard and Simpson Type Inequalities for DifferentiableP-GA-Functions

Authors: İmdat İşcan
Journal: International Journal of Analysis
 (2014)

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## Abstract

The author introduces the concept of theP-GA-functions, gives Hermite-Hadamard's inequalities for P-GA-functions, and defines a new identity. By using this identity, the author obtains new estimates on generalization of Hadamard and Simpson type inequalities for P-GA-functions. Some applications to special means of real numbers are also given.


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## Body

## 1. Introduction

Let real functionf be defined on some nonempty interval I of real line R. The function f is said to be convex on I if inequality
(1)f(tx+(1-t)y)≤tf(x)+(1-t)f(y)
holds for all x,y∈I and t∈[0,1].We recall that a functionf:I⊂R→R is said to be P-function on I or belong to the class P(I) if it is nonnegative and
(2)f(tx+(1-t)y)≤f(x)+f(y)
for all x,y∈I and t∈[0,1]. Note that P(I) contain all nonnegative convex and quasiconvex functions [1].The following inequalities are well known in the literature as Hermite-Hadamard inequality and Simpson inequality, respectively.Theorem 1.
Letf:I⊆R→R be a convex function defined on the interval I of real numbers and a,b∈I with a<b. The following double inequality holds:
(3)f(a+b2)≤1b-a∫abf(x)dx≤f(a)+f(b)2.Theorem 2.
Letf:[a,b]→R be a four times continuously differentiable mapping on (a,b) and ∥f(4)∥∞=sup⁡x∈(a,b)|f(4)(x)|<∞. Then the following inequality holds:
(4)|13[f(a)+f(b)2+2f(a+b2)]-1b-a∫abf(x)dx|≤12880∥f(4)∥∞(b-a)4.Definition 3 (see [2, 3]).
A functionf:I⊆(0,∞)→R is said to be GA-convex (geometric-arithmetically convex) if
(5)f(xty1-t)≤tf(x)+(1-t)f(y)
for all x,y∈I and t∈[0,1].In recent years, many authors have studied errors estimations for Hermite-Hadamard and Simpson inequalities; for refinements, counterparts, and generalization concerningP-functions and GA-convex, see [4–11].In this paper, the concept of theP-GA-function is introduced, Hermite-Hadamard’s inequalities for P-GA-functions are established, and a new identity for differentiable functions is defined. By using this identity, the author obtains a generalization of Hadamard and Simpson type inequalities for P-GA-functions.


## 2. Main Results

Letf:I⊆(0,∞)→R be a differentiable function on I∘, the interior of I; throughout this section we will take
(6)If(α,λ,a,b)=(1-λ)f(a1-αbα)+λ[αf(a)+(1-α)f(b)]-1ln⁡(b/a)∫abf(u)udu,
where a,b∈I with a<b and α,λ∈[0,1].Definition 4.
A functionf:I⊆(0,∞)→R is said to be P-GA-function (P-geometric-arithmetic function) on I if
(7)f(xty1-t)≤f(x)+f(y),
for any x,y∈I and t∈[0,1].Proposition 5.
Letf:I⊆(0,∞)→R. If f is P-function and nondecreasing, then f is P-GA-function on I.Proof.
This follows from(8)f(xty1-t)≤f(tx+(1-t)y)≤f(x)+f(y),
for all x,y∈I and t∈[0,1].Proposition 6.
Letf:I⊆(0,∞)→R. If f is P-GA-function and nonincreasing, then f is P-function on I.Proof.
The conclusion follows from(9)f(tx+(1-t)y)≤f(xty1-t)≤f(x)+f(y)
for all x,y∈I and t∈[0,1], respectively.Hermite-Hadamard’s inequalities can be represented forP-GA-functions as follows.Theorem 7.
Letf:I⊆(0,∞)→R be a function such that f∈L[a,b] (f is integrable on [a,b]), where a,b∈I with a<b. If f is a P-GA-function on [a,b], then the following inequalities hold:
(10)f(ab)≤2ln⁡(b/a)∫abf(u)udu≤2[f(a)+f(b)],
with α>0.Proof.
Sincef is a P-GA-function on [a,b], we have for all x,y∈[a,b] (with t=1/2 in inequality (7))
(11)f(xy)≤f(x)+f(y).
Choosing x=atb1-t,  y=bta1-t, we get
(12)f(ab)≤f(atb1-t)+f(bta1-t).
Integrating the resulting inequality with respect to t over [0,1], we obtain
(13)f(ab)≤∫01f(atb1-t)+f(bta1-t)dt=2ln⁡(b/a)∫abf(u)udu,
and the first inequality is proved.
For the proof of the second inequality in (10) we first note that if f is a P-GA-function, then, for t∈[0,1], it yields
(14)f(atb1-t)≤f(a)+f(b),f(bta1-t)≤f(a)+f(b).
By adding side to side these inequalities and taking square root we have
(15)f(atb1-t)+f(bta1-t)≤2[f(a)+f(b)],
and, integrating the resulting inequality with respect to t over [0,1], we obtain
(16)2ln⁡(b/a)∫abf(u)udu≤2[f(a)+f(b)].
The proof is completed.In order to prove our main results we need the following identity.Lemma 8.
Letf:I⊆(0,∞)→R be a differentiable function on I∘ such that f′∈L[a,b], where a,b∈I with a<b. Then for all x∈[a,b], λ∈[0,1], and α>0 one has
(17)If(α,λ,a,b)=(ln⁡ba){aα2∫01(t-λ)(ba)αtf′(a1-αtbαt)dt=(ln⁡ba)-b(1-α)2∫01(t-λ)(ab)(1-α)t=(ln⁡ba)-b(1-α)2×f′(a(1-α)tb1-(1-α)t)dt∫01(t-λ)(ba)αtf′(a1-αtbαt)}.Proof.
By integration by parts and changing the variable, we can state(18)a(ln⁡ba)α2∫01(t-λ)(ba)αtf′(a1-αtbαt)dt=α∫01(t-λ)df(a1-αtbαt)=α(t-λ)f(a1-αtbαt)|01-α∫01f(a1-αtbαt)dt=α(1-λ)f(a1-αbα)+αλf(a)-1ln⁡(b/a)∫aa1-αbαf(u)udu,
and similarly we get
(19)-b(ln⁡ba)(1-α)2∫01(t-λ)(ab)(1-α)t-b(ln⁡ba)(1-α)2×f′(a(1-α)tb1-(1-α)t)dt=(1-α)∫01(t-λ)df(a(1-α)tb1-(1-α)t)=(1-α)(t-λ)f(a(1-α)tb1-(1-α)t)|01-(1-α)∫01f(a(1-α)tb1-(1-α)t)dt=(1-α)(1-λ)f(a1-αbα)+(1-α)λf(b)-1ln⁡(b/a)∫a1-αbαbf(u)udu.
Adding the resulting identities we obtain the desired result.Theorem 9.
Letf:I⊂(0,∞)→R be a differentiable function on I∘ such that f′∈L[a,b], where a,b∈I∘ with a<b. If |f′|q is P-GA-function on [a,b] for some fixed q≥1, α,λ∈[0,1], then the following inequality holds:
(20)|If(α,λ,a,b)|≤(ln⁡ba)(λ2-λ+12)1-1/q(|f′(a)|q+|f′(b)|q)1/q×{aα2C1/q(λ,(ba)αq)+b(1-α)2C1/q(λ,(ab)(1-α)q)},
where
(21)C(λ,u)=1ln⁡2u[(u-λ(1+u))ln⁡u+2uλ-u-1].Proof.
Since|f′|q is P-GA-function on [a,b], for all t∈[0,1],
(22)|f′(a1-αtbαt)|q≤|f′(a)|q+|f′(b)|q,|f′(a(1-α)tb1-(1-α)t)|q≤|f′(a)|q+|f′(b)|q.
Hence, using Lemma 8 and power mean inequality, we get
(23)|If(α,λ,a,b)|≤(ln⁡ba)(∫01|t-λ|dt)1-1/q×{[|f′(a)|q+|f′(b)|q]dt∫01|t-λ|(ab)(1-α)qt)1/qaα2(∫01|t-λ|(ba)αqt×[|f′(a)|q+|f′(b)|q]dt∫01|t-λ|(ba)αqt)1/q+b(1-α)2×(∫01|t-λ|(ab)(1-α)qt×[|f′(a)|q+|f′(b)|q]dt∫01|t-λ|(ab)(1-α)qt)1/q}≤(ln⁡ba)(λ2-λ+12)1-1/q(|f′(a)|q+|f′(b)|q)1/q×{aα2(∫01|t-λ|(ba)αqtdt)1/q+b(1-α)2×(∫01|t-λ|(ab)(1-α)qtdt)1/q}≤(ln⁡ba)(λ2-λ+12)1-1/q(|f′(a)|q+|f′(b)|q)1/q×{aα2C1/q(λ,(ba)αq)+b(1-α)2C1/q(λ,(ab)(1-α)q)},
which completes the proof.Corollary 10.
Under the assumptions of Theorem9 with q=1, inequality (20) reduced to the following inequality:
(24)|If(α,λ,a,b)|≤(ln⁡ba)[|f′(a)|+|f′(b)|]×{aα2C(λ,(ba)α)+b(1-α)2C(λ,(ab)(1-α))}.Corollary 11.
Under the assumptions of Theorem9 with α=1/2, inequality (20) reduced to the following inequality:
(25)|∫abf(u)u(1-λ)f(ab)+λ[f(a)+f(b)2]-1ln⁡(b/a)∫abf(u)udu|≤14(ln⁡ba)(λ2-λ+12)1-1/q(|f′(a)|q+|f′(b)|q)1/q×{aC1/q(λ,(ba)q/2)+bC1/q(λ,(ab)q/2)}.
In particular, for λ=0, we get
(26)|f(ab)-1ln⁡(b/a)∫abf(u)udu|≤14(ln⁡ba)(12)1-1/q(|f′(a)|q+|f′(b)|q)1/q×{aC1/q(0,(ba)q/2)+bC1/q(0,(ab)q/2)}.
For λ=1, we get
(27)|f(a)+f(b)2-1ln⁡(b/a)∫abf(u)udu|≤14(ln⁡ba)(12)1-1/q(|f′(a)|q+|f′(b)|q)1/q×{aC1/q(1,(ba)q/2)+bC1/q(1,(ab)q/2)},
and, for λ=1/3,
(28)|13[f(a)+f(b)2+2f(ab)]-1ln⁡(b/a)∫abf(u)udu|≤14(ln⁡ba)(518)1-1/q(|f′(a)|q+|f′(b)|q)1/q×{aC1/q(13,(ba)q/2)+bC1/q(13,(ab)q/2)}.Theorem 12.
Letf:I⊂(0,∞)→R be a differentiable function on I∘ such that f′∈L[a,b], where a,b∈I∘ with a<b. If |f′|q is P-GA-function on [a,b] for some fixed q>1, α,λ∈[0,1], then the following inequality holds:
(29)|If(α,λ,a,b)|≤(ln⁡ba)(1p+1[λp+1+(1-λ)p+1])1/p×(|f′(a)|q+|f′(b)|q)1/q×{a1-αα2Lαq-1α-1/q(a,b)+bα(1-α)2L(1-α)q-11-α-1/q(a,b)},
where (1/p)+(1/q)=1 and Ln(a,b) is n-logarithmic mean defined with Ln(a,b):=((bn+1-an+1)/((n+1)(b-a)))1/n, n∈R∖{-1,0}.Proof.
Since|f′|q is P-GA-function on [a,b] and using Lemma 8 and Hölder inequality, we get
(30)|If(α,λ,a,b)|≤(ln⁡ba)(∫01|t-λ|pdt)1/p(|f′(a)|q+|f′(b)|q)1/q×{aα2(∫01(ba)αqtdt)1/q+b(1-α)2(∫01(ab)(1-α)qtdt)1/q}≤(ln⁡ba)(1p+1[λp+1+(1-λ)p+1])1/p×(|f′(a)|q+|f′(b)|q)1/q×{a1-αα2Lαq-1α-1/q(a,b)+bα(1-α)2L(1-α)q-11-α-1/q(a,b)}.
Here it is seen by simple computation that
(31)∫01|t-λ|pdt=1p+1[λp+1+(1-λ)p+1],∫01(ba)αqtdt=Lαq-1αq-1(a,b)aαq,∫01(ab)(1-α)qtdt=L(1-α)q-1(1-α)q-1(a,b)b(1-α)q.
Hence, the proof is completed.Corollary 13.
Under the assumptions of Theorem12 with α=1/2, inequality (29) reduced to the following inequality:
(32)(1-λ)f(ab)+λ[f(a)+f(b)2]-1ln⁡(b/a)∫abf(u)udu≤14(ln⁡ba)(1p+1[λp+1+(1-λ)p+1])1/p×(|f′(a)|q+|f′(b)|q)1/q×(a+b)Lq/2-1(q-2)/2q(a,b).
In particular, for λ=0, we get
(33)|f(ab)-1ln⁡(b/a)∫abf(u)udu|≤14(ln⁡ba)(1p+1)1/p(|f′(a)|q+|f′(b)|q)1/q×(a+b)Lq/2-1(q-2)/2q(a,b).
For λ=1, we get
(34)|f(a)+f(b)2-1ln⁡(b/a)∫abf(u)udu|≤14(ln⁡ba)(1p+1)1/p(|f′(a)|q+|f′(b)|q)1/q×(a+b)Lq/2-1(q-2)/2q(a,b),
and, for λ=1/3, we get
(35)|13[f(a)+f(b)2+2f(ab)]-1ln⁡(b/a)∫abf(u)udu|≤14(ln⁡ba)(1+2p+1(p+1)3p+1)1/p(|f′(a)|q+|f′(b)|q)1/q×(a+b)Lq/2-1(q-2)/2q(a,b).


## 3. Application to Special Means

Let us recall the following special means of two nonnegative numbersa, b with b>a:(1)
the arithmetic mean(36)A=A(a,b):=a+b2,(2)
the weighted arithmetic mean(37)Aα=Aα(a,b):=αa+(1-α)b,α∈[0,1],(3)
the geometric mean(38)G=G(a,b):=a+b2,(4)
the weighted geometric mean(39)Gα=Gα(a,b):=aαb1-α,α∈[0,1],(5)
the logarithmic mean(40)L=L(a,b):=b-aln⁡b-ln⁡a,(6)
then-logarithmic mean
(41)Ln=Ln(a,b):=(bn+1-an+1(n+1)(b-a))1/n,n∈R∖{-1,0}.Proposition 14.
Forb>a>0, n∈N, n≥2, and q≥1, one has
(42)|(1-λ)G1-αn(a,b)+λAα(an,bn)-Lnn(a,b)|≤n(ln⁡ba)(λ2-λ+12)1-1/q(a(n-1)q+b(n-1)q)1/q×{aα2C1/q(λ,(ba)αq)+b(1-α)2C1/q(λ,(ab)(1-α)q)},
where C is defined as in (21).Proof.
Letf(x)=xn, x>0, n≥2, andq≥1.Proposition 15.
Forb>a>0, n∈N, n≥2, and q>1, one has
(43)|(1-λ)G1-αn(a,b)+λAα(an,bn)-Lnn(a,b)|≤n(ln⁡ba)(1p+1[λp+1+(1-λ)p+1])1/p×(a(n-1)q+b(n-1)q)1/q×{a1-αα2Lαq-1α-1/q(a,b)+bα(1-α)2L(1-α)q-11-α-1/q(a,b)}.Proof.
Letf(x)=xn, x>0, n≥2, and q>1.



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