# Generalization of Some Simpson-Like Type Inequalities via Differentiables-Convex  Mappings in the Second Sense

Authors: Jaekeun Park
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

The author obtained new generalizations and refinements of some
inequalities based on differentiables-convex mappings in the second sense. Also,
some applications to special means of real numbers are given.


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## Body

## 1. Introduction

Recall that the functionf:[0,b]→ℝ, b>0 is said to be s-convex in the second sense for s∈[0,1] if (1.1)f(tx+(1-t)y)≤tsf(x)+(1-t)sf(y),
for all x,y∈[0,b] and t∈[0,1] [1–5].In (1.1), if we let s=1, f:𝕀⊂[0,∞)→ℝ is said to be a convex mapping on an interval 𝕀[1].Let us denote the set ofs-convex mappings in the second sense on 𝕀 by Ks2(𝕀).For some further properties of thes-convex mappings, see [1–3, 6]. In recent,  M. Z. Sarikaya et al. [4], and U. S. Kirmaci et al. [7] established a more general result of the Hermite-Hadamard inequalities.For recent years many authors have established error estimations for the Simpson’s inequality: for refinements, counterparts, generalizations, and new Simpson's type inequalities, see [1, 4, 6, 8].S. S. Dragomir et al. [9], and M. Alomari et al. [8] proved the following developments on Simpson's inequality for which the remainder is expressed in terms of lower derivatives than the twice.In the sequel, denote the interior of an interval𝕀 by 𝕀0.Theorem 1.1.
Letf:𝕀⊂[0,∞)→ℝ be an absolutely continuous mapping on [a,b] such that f′∈Lp([a,b]), where a,b∈𝕀 with a<b. Then the following inequality holds:
(1.2)|13{f(a)+f(b)2+2f(a+b2)}-1b-a∫abf(x)dx|≤(b-a)1/q6{2q+1+13(q+1)}1/q‖f′‖p.In this article, the author gives some generalized Simpson’s type inequalities based ons-convex mappings in the second sense by using the following lemma.


## 2. Generalization of Inequalities Based on s-Convex Mappings

In this article, for the simplicity of the notation, let(2.1)Sab(f)(h,n)=1n{f(a)+(n-2)f(hb+(1-h)a)+f(b)}-1b-a∫abf(x)dx,for h∈(0,1) with 1/n≤h≤(n-1)/n for any integer n≥2.In order to generalize the classical Simpson-like type inequalities, we need the following lemma [1, 6].Lemma 2.1.
Letf:[0,b]→ℝ, b>0 be a differentiable mapping on 𝕀0 such that f′∈L[a,b], where a,b∈𝕀 with a<b and [a,b]⊂[0,b]. If f′∈L1([a,b]), then, for h∈(0,1) with 1/n≤h≤(n-1)/n for any n≥2 the following equality holds:
(2.2)Sab(f)(h,n)=(b-a)∫01p(t,h)f′(tb+(1-t)a)dt,
for each t∈[0,1], where
(2.3)p(t,h)={t-1n,t∈[0,h],t-n-1n,t∈(h,1].Proof.
By the integration by parts, we have(2.4)∫0h(t-1n)f′(tb+(1-t)a)dt+∫h1(t-n-1n)f′(tb+(1-t)a)dt=1b-a{n-2nf(hb+(1-h)a)+1n(f(a)+f(b))}-1(b-a)2∫abf(x)dx,
which completes the proof.Theorem 2.2.
Letf:[0,b]→ℝ, b>0 be a differentiable mapping on 𝕀0 such that f′∈L[a,b], where a,b∈𝕀 with a<b and [a,b]⊂[0,b]. If |f′|∈Ks2([a,b]), for some s∈(0,1], then for h∈(0,1) with 1/n≤h≤n-1/n for any n≥2 the following inequality holds:
(2.5)|Sab(f)(h,n)|≤(b-a){λ11|f′(b)|+μ11|f′(a)|},
where
(2.6)λ11=2+2(n-1)s+2ns+2(s+1)(s+2)+2+s-nn(s+1)(s+2)+hs+1(s(2h-1)+2(h-1))(s+1)(s+2),μ11=2+2(n-1)s+2ns+2(s+1)(s+2)+2+s-nn(s+1)(s+2)+(1-h)s+1{s(1-2h)-2h}(s+1)(s+2).Proof.
From Lemma2.1 and since |f′| is s-convex on [a,b], by using Hölder integral inequality, we have
(2.7)|Sab(f)(h,n)|≤(b-a)∫01/n(1n-t)∣(ts|f′(b)|+(1-t)s|f′(a)|)dt+(b-a)∫1/nh(t-1n)(ts|f′(b)|+(1-t)s|f′(a)|)dt+(b-a)∫h(n-1)/n(n-1n-t)(ts|f′(b)|+(1-t)s|f′(a)|)dt+(b-a)∫(n-1)/n1(t-n-1n)(ts|f′(b)|+(1-t)s|f′(a)|)dt=(b-a){λ11|f′(b)|+μ11|f′(a)|},
which implies the theorem.Corollary 2.3.
In Theorem2.2, one has:
(i)(2.8)|Sab(f)(12,n)|≤(b-a){2+2(n-1)s+2+ns+1(s-n+2-2-(s+1)n)ns+2(s+1)(s+2)}{|f′(a)|+|f′(b)|},
(ii)(2.9)|Sab(f)(12,6)|≤(b-a){6-s(1+5s+2-3s+2)+3(s-4)18(s+1)(s+2)}{|f′(a)|+|f′(b)|},
which implies that Corollary 2.3 is a generalization of Theorem 1.1.Theorem 2.4.
Letf:[0,b]→ℝ, b>0 be a differentiable mapping on 𝕀0 such that f′∈L[a,b], where a,b∈𝕀 with a<b and [a,b]⊂[0,b]. If |f′|q∈Ks2[a,b], for some fixed s∈(0,1] and q>1 with 1/p+1/q=1, then for h∈(0,1) with 1/n≤h≤(n-1)/n for any n≥2 the following inequality holds:
(2.10)|Sab(f)(h,n)|≤(b-a){1+(hn-1)p+1np+1(p+1)}1/p(hs+1)1/q×{|f′(hb+(1-h)a)|q+|f′(a)|q}1/q+(b-a){1+(n-hn-1)p+1np+1(p+1)}1/p(1-hs+1)1/q×{|f′(b)|q+|f′(hb+(1-h)a)|q}1/q.Proof.
From Lemma2.1, using the Hölder inequality we get
(2.11)|Sab(f)(h,n)|≤(b-a)(∫0h|t-1n|pdt)1/p(∫0h|f′(tb+(1-t)a)|qdt)1/q+(b-a)(∫h1|t-n-1n|pdt)1/p(∫h1|f(tb+(1-t)a)|qdt)1/q.
Since |f′|q∈Ks2([a,b]) for a fixed s∈(0,1], we have(a)
(2.12)∫0h|f′(tb+(1-t)a)|qdt≤(hs+1){|f′(hb+(1-h)a)|q+|f′(a)|q},(b)
(2.13)∫h1|f′(tb+(1-t)a)|qdt≤(1-hs+1){|f′(b)|q+|f′(hb+(1-h)a)|q}.
By (2.11) and (2.12), the assertion (2.10) holds.Corollary 2.5.
In Theorem2.4, letting n=6 and h=1/2, one has
(2.14)|Sab(f)(12,6)|≤(b-a)(2p+1+16p+1(p+1))1/p(12(s+1))1/q×[{|f′(a)|q+|f′(a+b2)|q}1/q+{|f′(a+b2)|q+|f′(b)|q}1/q].Theorem 2.6.
Letf:[0,b]→ℝ, b>0 be a differentiable mapping on 𝕀0 such that f′∈L[a,b], where a,b∈𝕀 with a<b and [a,b]⊂[0,b]. If |f′|q∈Ks2([a,b]), for some fixed s∈(0,1] and q>1 with 1/p+1/q=1, then for h∈(0,1) with 1/n≤h≤(n-1)/n for any n≥2 the following inequality holds:
(2.15)|Sab(f)(h,n)|≤(b-a){1np+1(p+1)}1/p(1s+1)1/q×[{(1-h)s+1}1/q{1+(hn-1)p+1}1/p{(hs+1|f′(b)|q+(1-(1-h)s+1)|f′(a)|q}1/q+{1+(n-nh-1)p+1}1/p×{((1-hs+1)|f′(b)|q+(1-h)s+1|f′(a)|q}1/q].Proof.
Note that(2.16)(a)∫0h|f′(tb+(1-t)a)|qdt≤(hs+1s+1)|f′(b)|q+(1-(1-h)s+1s+1)|f′(a)|q,(b)∫h1|f′(tb+(1-t)a)|qdt≤(1-hs+1s+1)|f′(b)|q+((1-h)s+1s+1)|f′(a)|q.
By (2.11) and (2.16), the assertion (2.15) of this theorem holds.Corollary 2.7.
In Theorem2.6, letting h=1/2, then one has
(2.17)|Sab(f)(12,n)|≤(b-a){2p+1+(n-2)p+12p+1np+1(p+1)}1/p(12s+1(s+1))1/q×[{|f′(b)|q+(2s+1-1)|f′(a)|q}1/q+{(2s+1-1)|f′(b)|q+|f′(a)|q}1/q],
which implies that Theorem 2.6 is a generalization of Theorem 1.1.Theorem 2.8.
Letf:[0,b]→ℝ, b>0 be a differentiable mapping on 𝕀0 such that f′∈L[a,b], where a,b∈𝕀 with a<b and [a,b]⊂[0,b]. If |f′|q∈Ks2([a,b]), for some fixed s∈(0,1] and q≥1 with 1/p+1/q=1, then for h∈(0,1) with 1/n≤h≤(n-1)/n for any n≥2 the following inequality holds:
(2.18)|Sab(f)(h,n)|≤(b-a){1+(nh-1)22n2}1/p{λ21|f′(b)|q+ν21|f′(a)|q}1/q+(b-a){1+(nh-n+1)22n2}1/p{λ22|f′(b)|q+ν22|f′(a)|q}1/q,
where
(2.19)λ21=2ns+2(s+1)(s+2)+hs+1(-2-s+hn+hns)n(s+1)(s+2),ν21=2(n-1)s+2ns+2(s+1)(s+2)+(s+2-n)(1+(1-h)s+1)n(s+1)(s+2)-h(1-h)s+1s+2,λ22=2(n-1)s+2ns+2(s+1)(s+2)+(s+2-n)(1+hs+1)n(s+1)(s+2)-hs+1(1-h)s+2,ν22=2ns+2(s+1)(s+2)+(1-h)s+1(-2-s+n-hn+ns-hns)n(s+1)(s+2).Proof.
Suppose thatq≥1. From Lemma 2.1, using the power mean inequality one has
(2.20)|Sab(f)(h,n)|≤(b-a)[(∫0h|t-1n|dt)1/p(∫0h|t-1n||f′(tb+(1-t)a)|qdt)1/q+(∫h1|t-n-1n|dt)1/p(∫h1|t-n-1n||f′(tb+(1-t)a)|qdt)1/q].
Since|f′| is s-convex on [a,b], we have
(2.21)∫0h|t-1n||f′(tb+(1-t)a)|qdt≤λ21|f′(b)|q+ν21|f′(a)|q∫h1|t-n-1n||f(tb+(1-t)a)|qdt≤λ22|f′(b)|q+ν22|f′(a)|q.
By the above facts (2.20) and (2.21), the assertion (2.18) in this theorem is proved.Corollary 2.9.
In Theorem2.8, letting h=1/2, one has
(2.22)λ21=ν22=2ns+2(s+1)(s+2)+(n-2)(s+1)-22s+2n(s+1)(s+2),λ22=ν21=ns+22-(s+2)(s+1)+2(n-1)s+2ns+2(s+1)(s+2)+(s-n+2)(2s+1+1)2s+1n(s+1)(s+2),
which implies that
(2.23)|Sab(f)(12,n)|≤(b-a)(18-12n+1n2)1/p×[{λ21|f′(b)|q+λ22|f′(a)|q}1/q+{λ22|f′(b)|q+λ21|f′(a)|q}1/q].
Especially, in Theorem2.8, letting h=1/2 and m=1, one has
(2.24)|Sab(f)(12,n)|≤(b-a)(18-12n+1n2)1/p×{(λ21|f′(b)|q+λ22|f′(a)|q)1/q+(λ22|f′(b)|q+λ21|f′(a)|q)1/q}.


## 3. Applications to Special Means

We now consider the applications of our theorems to the followings special means.(a) The arithmetic mean:A(a,b)=(a+b)/2,a,b≥0.(b) Thep-logarithmic mean:(3.1)Lp(a,b)={[bp+1-ap+1(p+1)(b-a)]1/p,ifa≠b,a,ifa=b,forp∈ℝ∖{-1,0} and a,b>0.Now, using the results of Section2, some new inequalities are derived for the following means:(1.1) Let f:[a,b]→ℝ,(0<a<b),f(x)=xs,s∈(0,1].(a) In Theorem2.2,(i) ifh=1/2 and n≥2, then we get
(3.2)|1n[2A(as,bs)+(n-2)As(a,b)]-Lss(a,b)|≤(b-a)2s{2+2(n-1)s+2+ns+1(2+s-n-2-(s+1)n)ns+2(s+1)(s+2)}A(as-1,bs-1),and,(ii) ifh=1/2 and n=6, then we have
(3.3)|13[A(as,bs)+2As(a,b)]-Lss(a,b)|≤(b-a)s{6-s+5s+26-s+3(s-4-2-s3)9(s+1)(s+2)}A(as-1,bs-1).(b) In Theorem2.4,(i) ifh=1/2, n≥2 and q>1 then we get:
(3.4)|1n[2A(as,bs)+(n-2)As(a,b)]-Lss(a,b)|≤b-a2{1+((n/2)-1)p+1np+1(p+1)}1/p(sq2(s+1))1/q×[{A(s-1)q(A(a,b))+a(s-1)q)}1/q+{A(s-1)q(A(a,b))+b(s-1)q)}1/q],and(ii) ifh=1/2, n=6 and q>1, then we have (3.5)|13[A(as,bs)+2As(a,b)]-Lss(a,b)|≤b-a12{1+2p+13(p+1)}1/p(sq(s+1))1/q×[{A(s-1)q(A(a,b))+a(s-1)q)}1/q+{A(s-1)q(A(a,b))+b(s-1)q)}1/q],(c)  In Theorem2.6,(i) ifh=1/2, n≥2 and q>1 then we get
(3.6)|1n[2A(as,bs)+(n-2)As(a,b)]-Lss(a,b)|≤(b-a){1+((n/2)-1)p+1np+1(p+1)}1/p{sq2s+1(s+1)}1/q×[{b(s-1)q+(2s+1-1)a(s-1)q}1/q+{(2s+1-1)b(s-1)q+a(s-1)q}1/q],and(ii) ifh=1/2, n=6 and q>1, then we have (3.7)|13[A(as,bs)+2As(a,b)]-Lss(a,b)|≤(b-a){1+2p+16p+1(p+1)}1/p{sq2s+1(s+1)}1/q×[{b(s-1)q+(2s+1-1)a(s-1)q}1/q+{(2s+1-1)b(s-1)q+a(s-1)q}1/q].
In Theorem 2.8,(i) ifh=1/2, n≥2 and q≥1 then we get(3.8)|1n[2A(as,bs)+(n-2)As(a,b)]-Lss(a,b)|≤(b-a){1+((n/2)-1)22n2}1/p×[{λ21′b(s-1)q+λ22′a(s-1)q}1/q+{λ22′b(s-1)q+λ21′b(s-1)q}1/q],where(3.9)λ21′=2ns+2(s+1)(s+2)+(ns/2)+(n/2)-s-2n2s+1(s+1)(s+2),λ22′=2(n-1)s+2ns+2(s+1)(s+2)+(s-n+2)(2s+1+1)n2s+1(s+1)(s+2)-12s+2(s+2),and(ii) ifh=1/2, n=6 and q≥1, then we have (3.10)|13{A(as,bs)+2As(a,b)}-Lss(a,b)|≤(b-a)(572)1/p[{λ21′′b(s-1)q+λ22′′a(s-1)q}1/q+{λ22′′b(s-1)q+λ21′′b(s-1)q}1/q],where(3.11)λ21′′=2+3s+1(2s+1)6s+2(s+1)(s+2),λ22′′=2⋅5s+2+3s+1(2s+1-2)s-3s+1(2s+3+7)6s+2(s+1)(s+2).(2.2) Let f:[a,b]→ℝ,(0<a<b),f(x)=1/xs,s∈(0,1].(a) In Theorem2.2,(i) ifh=1/2 and n≥2, then we get
(3.12)|1n[2A(a-s,b-s)+(n-2)A-s(a,b)]-L-s-s(a,b)|≤(b-a)2s{2s+2(1+(n-1)s+2)+2s+1ns+1(2+s-n)-ns+2ns+22s+1(s+1)(s+2)}×A(a-(s+1),b-(s+1)),and(ii) ifh=1/2 and n=6, then we have (3.13)|13[A(a-s,b-s)+2A-s(a,b)]-L-s-s(a,b)|≤23(b-a)s{1+5s+2+3(s-4)6s-3s+26s+2(s+1)(s+2)}A(a-(s+1),b-(s+1)).(b) In Theorem2.4,(i) ifh=1/2 and n≥2, then we get (3.14)|1n[2A(a-s,b-s)+(n-2)A-s(a,b)]-L-s-s(a,b)|≤(b-a){1+((n/2)-1)p+1np+1(p+1)}1/p{sq2(s+1)}1/q×[{A-(s+1)q(a,b)+a-(s+1)q}1/q+{A-(s+1)q(a,b)+b-(s+1)q}1/q],and(ii) ifh=1/2 and n=6, then we have (3.15)|13[A(a-s,b-s)+2A-s(a,b)]-L-s-s(a,b)|≤(b-a){1+2p+16p+1(p+1)}1/p{sq(s+1)}1/q×{A1/q(A-(s+1)q(a,b),a-(s+1)q)+A1/q(A-(s+1)q(a,b),b-(s+1)q)}.(c) In Theorem2.6,(i) ifh=1/2 and n≥2, then we get(3.16)|1n[2A(a-s,b-s)+(n-2)A-s(a,b)]-L-s-s(a,b)|≤(b-a){1np+1(p+1)}1/p{sq2s+1(s+1)}1/q{1+(n2-1)p+1}1/p×[{b-(s+1)q+(2s+1-1)a-(s+1)q}1/q+{(2s+1-1)b-(s+1)q+a-(s+1)q}1/q],and(ii) ifh=1/2 and n=6, then we have (3.17)|13[A(a-s,b-s)+2A-s(a,b)]-L-s-s(a,b)|≤(b-a){1+2p+16p+1(p+1)}1/p{sq2s+1(s+1)}1/q×[{b-(s+1)q+(2s+1-1)a-(s+1)q}1/q+{(2s+1-1)b-(s+1)q+a-(s+1)q}1/q].(d) In Theorem2.8,(i) ifh=1/2, n≥2 and q≥1 then we get(3.18)|1n[2A(a-s,b-s)+2-s(n-2)A-s(hb,(1-h)a)]-L-s-s(a,b)|≤(b-a)s{(n-2)2+48n2}1/p×[{λ31′b(s-1)q+λ32′a(s-1)q}1/q+{λ32′b(s-1)q+λ31′b(s-1)q}1/q],where(3.19)λ31′=2ns+2(s+1)(s+2)+-2-s+n/2+ns/2n2s+2(s+1)(s+2),λ32′=2(n-1)s+2ns+2(s+1)(s+2)+(s-n+2)(2s+1+1)n2s+1(s+1)(s+2)-12s+2(s+2),and(ii) ifh=1/2, n=6 and q≥1, then we have (3.20)|13[A(a-s,b-s)+2A-s(a,b)]-L-s-s(a,b)|≤(b-a)s{572}1/p×[{λ31′′bq(s-1)+λ32′′aq(s-1)}1/q+{λ32′′bq(s-1)+λ31′′bq(s-1)}1/q],where(3.21)λ31′′=12+3s+2(2s+1)6s+3(s+1)(s+2),λ32′′=2⋅5s+2+3s+1(2s+1-4)s+3s+1(1-2s+3)6s+2(s+1)(s+2).



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