# New Inequalities for Stronglyr-Convex Functions

Authors: Huriye Kadakal
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this study, firstly we introduce a new concept called “stronglyr-convex function.” After that we establish Hermite-Hadamard-like inequalities for this class of functions. Moreover, by using an integral identity together with some well known integral inequalities, we establish several new inequalities for n-times differentiable strongly r-convex functions. In special cases, the results obtained coincide with the well-known results in the literature.


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## Body

## 1. Introduction

A functionf:I⊆R→R is said to be convex if the inequality (1)ftx+1-ty≤tfx+1-tfy is valid for all x,y∈I and t∈[0,1]. If this inequality reverses, then the function f is said to be concave on interval I≠∅. This definition is well known in the literature. Convexity theory has appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences. Many articles have been written by a number of mathematicians on convex functions and inequalities for their different classes, using, for example, the last articles [1–6] and the references in these papers.Letf:a,b→R be a convex function; then the inequality(2)fa+b2≤1b-a∫abfxdx≤fa+fb2 is known as the Hermite-Hadamard inequality (see [7] for more information). Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1, 4, 8]). In [9], the first author obtained a new refinement of the Hermite-Hadamard inequality for convex functions. The Hermite-Hadamard inequality was generalized in [10] to an r-convex positive function which is defined on an interval a,b.Definition 1.
A positive functionf:a,b⊆R→R is called r-convex function on a,b, if, for each x,y∈a,b and t∈0,1, (3)ftx+1-ty≤tfrx+1-tfry1/r,r≠0,fxtfy1-t,r=0.If the equality is reversed, then the function f is said to be r-concave.It is obvious that0-convex functions are simply log-convex functions, 1-convex functions are ordinary convex functions, and -1-convex functions are arithmetically harmonically convex. One should note that if the function f is r-convex on a,b, then the function fr is a convex function for r>0 and fr is a concave function for r<0. We note that if the functions f and g are convex and g is increasing, then gof is convex; moreover, since f=exp⁡log⁡f, it follows that a log-convex function is convex.The definition ofr-convexity naturally complements the concept of r-concavity, in which the inequality is reversed [11] and plays an important role in statistics.It is easily seen that iff is r-convex on a,b, (4)fra+b2≤1b-a∫abfrxdx≤fra+frb2,r>0(5)fra+b2≥1b-a∫abfrxdx≥fra+frb2,r<0Some refinements of the Hadamard inequality forr-convex functions could be found in [12–16]. In [14], the authors showed that if the function f is r-convex in a,b and 0<r≤1, then (6)1b-a∫abfxdx≤rr+1fra+frb1/r.Theorem 2 (see [17]).
Suppose thatf is a positive r-convex function on a,b. Then (7)1b-a∫abftdt≤Lrfa,fb.If the functionf is a positive r-concave function, then the inequality is reversed, where (8)Lrfa,fb=rr+1fr+1a-fr+1bfra-frb,fa≠fbfa,fa=fb.Definition 3.
LetI⊂R be an interval and c be a positive number. A function f:I⊂R→R is called strongly convex with modulus c if (9)ftx+1-ty≤tfx+1-tfy-ct1-tx-y2for all x,y∈I and t∈0,1.In this definition, if we takec=0, we get the definition of convexity in the classical sense. Strongly convex functions have been introduced by Polyak [18], and they play an important role in optimization theory and mathematical economics. Since strong convexity is a strengthening of the notion of convexity, some properties of strongly convex functions are just “stronger versions” of known properties of convex functions. For more information on strongly convex functions, see [19–21] and references therein.Lemma 4.
Leta≥0, b≥0. Then a+bλ≤aλ+bλ,0<λ≤1.Lemma 5 (Minkowski’s integral inequality).
Letf,g∈Lpa,b. If p>1, then Minkowski’s integral inequality states that(10)∫abfx+gxpdx1/p≤∫abfxpdx1/p+∫abgxpdx1/p.Let0<a<b, throughout this paper we will use(11)Aa,b=a+b2,Ga,b=abLpa,b=bp+1-ap+1p+1b-a1/p,a≠b,p∈R,p≠-1,0for the arithmetic, geometric, and generalized logarithmic mean, respectively. Also for shortness we will use the following notation: (12)Ia,b,n,f=∑k=0n-1-1kfkbbk+1-fkaak+1k+1!-∫abfxdxwhere an empty sum is understood to be nil.


## 2. Main Results

In this section we introduce a new concept, which is called stronglyr-convex function, as follows.Definition 6.
A positive functionf:I⊆R→R is called strongly r-convex function with modulus c on a,b, if, for each x,y∈a,b and t∈0,1, (13)ftx+1-ty≤tfrx+1-tfry1/r-ct1-tx-y2,r≠0,In this definition, if we takec=0, we get the definition of r-convexity in the classical sense.Theorem 7.
Letf:0,∞→R be strongly r-convex function with modulus c on a,b with a<b. Then the following inequality holds for 0<r≤1:(14)1b-a∫abfxdx≤rr+1far+fbr1/r-c6a-b2Proof.
Since the functionf is strongly r-convex function and r>0, we have(15)fta+1-tb≤tfra+1-tfrb1/r-ct1-ta-b2for all t∈0,1. It is easy to observe that(16)1b-a∫abfxdx=∫01fta+1-tbdt≤∫01tfar+1-tfbr1/r-ct1-ta-b2dt.Using Minkowski’s integral inequality, we obtain(17)∫01tfar+1-tfbr1/r-ct1-ta-b2dt=∫01tfar+1-tfbr1/rdt-c∫01t1-ta-b2dt≤∫01t1/rfadtr+∫011-t1/rfbdtr1/r-ca-b2∫01t1-tdt=rr+1far+rr+1fbr1/r-c6a-b2.Thus(18)1b-a∫abfxdx≤rr+1far+fbr1/r-c6a-b2.This proof is complete.Theorem 8.
Suppose thatf is a positive strongly r-convex function with modulus c on a,b. Then (19)1b-a∫abftdt≤Lrfa,fb-c6a-b2,where (20)Lrfa,fb=rr+1fr+1a-fr+1bfra-frb,fa≠fbfa,fa=fb.Proof.
Firstly, assume thatf(a)≠f(b). By (13) we have(21)∫abfxdx=b-a∫01fta+1-tbdx≤b-a∫01tfra+1-tfrb1/r-ct1-ta-b2dt=b-a∫01tfra+1-tfrb1/rdt-ca-b2∫01t1-tdt=b-a∫frafrbu1/rfrb-fradu-ca-b2∫01t1-tdt=b-a1frb-frau1/r+11/r+1frafrb-ca-b2t22-t3301=b-arr+1fr+1b-fr+1afrb-fra-c6a-b2=b-aLrfa,fb-c6a-b2.Secondly, for f(a)=f(b), we get(22)∫abfxdx≤b-a∫01tfra+1-tfrb1/r-ct1-ta-b2dt=b-a∫01fa-ct1-ta-b2dt=b-afa-c6a-b2=b-aLrfa,fa-c6a-b2.This proof is complete.We will use the following lemma for obtaining our main results.Lemma 9 (see [6]).
Letf:I⊆R→R be n-times differentiable mapping on I∘ for n∈N and fn∈L[a,b], where a,b∈I∘ with a<b; we have the identity (23)Ia,b,n,f=-1n+1n!∫abxnfnxdx.where an empty sum is understood to be nil.We note that the authors obtained several new integeral inequalities forn-times differentiable log-convex, s-convex functions in the first sense, strongly convex, r-Convex and r-Concave, and convex and concave functions using the above lemma (see [5, 6, 22–24]). In this paper, we consider n-times differentiable strongly r-convex function and establish several new inequalities for this class of functions. Obtained results in this paper coincide with the results of papers ([6, 23, 24]).Theorem 10.
Forn in N, let f:I⊂0,∞→R be n-times differentiable function on I∘, r>0 and a,b∈I∘ with a<b. If f(n)∈L[a,b] and f(n)q for q>1 is strongly r-convex function with modulus c on [a,b], then the following inequality holds: (24)Ia,b,n,f≤b-an!Lnpna,bL1/r1/rfnaqr,fnbqr-cb-a261/qProof.
Iff(n)q for q>1 is strongly r-convex function on [a,b] and r>0, using Lemma 9, the Hölder integral inequality and the inequality(25)fnxq=fnx-ab-ab+b-xb-aaq≤x-ab-afnbqr+b-xb-afnaqr1/r-cb-xx-a,we have (26)
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1
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1
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n
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1
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.This completes the proof of theorem.Remark 11.
The following results are remarkable for Theorem10.
(i) The results obtained in this paper reduce to the results of [24] in case of c=0.
(ii) The results obtained in this paper reduce to the results of [23] in case of r=1.
(iii) The results obtained in this paper reduce to the results of [6] in case of r=1 and c=0.Corollary 12.
Under the conditions Theorem10 for n=1 we have the following inequality: (27)fbb-faab-a-1b-a∫abfxdx≤Lpa,bL1/r1/rf′aqr,f′bqr-cb-a261/q.Theorem 13.
Forn in N, let f:I⊂0,∞→R be n-times differentiable function on I∘, r>0 and a,b∈I∘ with a<b. If f(n)∈L[a,b] and f(n)q for q≥1 is strongly r-convex function with modulus c on [a,b], then the following inequality holds: (28)Ia,b,n,f≤b-a1-1/q-1/qrn!Lnnq-1/qa,b×C1fnbq+C2fnaq+cb-a1+1/r×Ln+2n+2a,b-2Aa,bLn+1n+1a,b+G2a,bLnna,b1/q,r≥1C1rfnbqr+C2rfnaqr1/r+cb-a1+1/r×Ln+2n+2a,b-2Aa,bLn+1n+1a,b+G2a,bLnna,b1/q,r≤1where (29)C1=C1a,b,r,n=∫abxnx-a1/rdx,C2=C2a,b,r,n=∫abxnb-x1/rdx.Proof.
From Lemma9 and Power-Mean integral inequality, we get (30)Ia,b,n,f≤1n!∫abxnfnxdx≤1n!∫abxndx1-1/q∫abxnfnxqdx1/q≤1n!∫abxndx1-1/q∫abxnx-ab-afnbqr+b-xb-afnaqr1/r-cb-xx-adx1/q.Here, using Lemma 4 we obtain, respectively, the following.
Forr≥1(31)
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1
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2
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1
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.This completes the proof of theorem.Remark 14.
The following results are remarkable for Theorem13.
(i) The results obtained in this paper reduce to the results of [24] in case of c=0.
(ii) The results obtained in this paper reduce to the results of [23] in case of r=1.
(iii) The results obtained in this paper reduce to the results of [6] in case of r=1 and c=0.Corollary 15.
Under the conditions Theorem13 for n=1 we have the following inequalities: (33)Ja,b,f≤Aq-1/qa,b×f′bq-f′aqrb-a2r+1+af′bq-bf′aqrr+1+cL33a,b-2Aa,bL22a,b+G2a,bAa,b1/q,r≥1a+br2+brr+12r+1rf′bqr+a+br2+arr+12r+1rf′aqr1/r+cL33a,b-2Aa,bL22a,b+G2a,bAa,b1/q,r≤1where Ja,b,f=Ia,b,1,f/b-a.Corollary 16.
Under the conditions Theorem13 for q=1 we have the following inequalities:(34)Ia,b,n,f≤1n!b-a-1/r×C1fnb+C2fna+cb-a1+1/r×Ln+2n+2a,b-2Aa,bLn+1n+1a,b+G2a,bLnna,b,r≥1C1rfnbr+C2rfnar1/r+cb-a1+1/r×Ln+2n+2a,b-2Aa,bLn+1n+1a,b+G2a,bLnna,b,r≤1Theorem 17.
Forn in N, let f:I⊂0,∞→R be n-times differentiable function on I∘, r>0 and a,b∈I∘ with a<b. If f(n)∈L[a,b] and f(n)q for q>1 is strongly r-convex function with modulus c on [a,b], then the following inequality holds:(35)Ia,b,n,f≤b-a1/p-1/qrn!×fnbqD1+fnaqD2+cb-a1+1/qr×Lnq+2nq+2a,b-2Aa,bLnq+1nq+1a,b+G2a,bLnqnqa,b1/q,r≥1fnbqrD1r+fnaqrD2r+cb-a1+1/qr×Lnq+2nq+2a,b-2Aa,bLnq+1nq+1a,b+G2a,bLnqnqa,b1/qr,r≤1where(36)D1=D1a,b,r,n,q=∫abxnqx-a1/rdxD2=D2a,b,r,n,q=∫abxnqb-x1/rdx.Proof.
Sincef(n)q for q>1 is strongly r-convex function on [a,b], using Lemma 9 and the Hölder integral inequality, we have the following inequality: (37)
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Forr≥1, (38)
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a
b
x
n
q
b
-
x
b
-
a
1
/
r
d
x
-
c
b
-
a
1
/
q
r
∫
a
b
x
n
q
b
-
x
x
-
a
d
x
=
1
n
!
b
-
a
1
/
p
-
1
/
q
r
f
n
b
q
D
1
+
f
n
a
q
D
2
+
c
b
-
a
1
+
1
/
q
r
×
L
n
q
+
2
n
q
+
2
a
,
b
-
2
A
a
,
b
L
n
q
+
1
n
q
+
1
a
,
b
+
G
2
a
,
b
L
n
q
n
q
a
,
b
1
/
q
.For r≤1, using Minkowski’s integral inequality, we get (39)
I
a
,
b
,
n
,
f
≤
1
n
!
b
-
a
1
/
p
-
1
/
q
r
∫
a
b
x
n
q
r
x
-
a
f
n
b
q
r
+
x
n
q
r
b
-
x
f
n
a
q
r
1
/
r
-
c
b
-
a
1
/
q
r
∫
a
b
x
n
q
b
-
x
x
-
a
d
x
1
/
q
=
1
n
!
b
-
a
1
/
p
-
1
/
q
r
f
n
b
q
∫
a
b
x
n
q
x
-
a
1
/
r
d
x
r
+
f
n
a
q
∫
a
b
x
n
q
b
-
x
1
/
r
d
x
r
-
c
b
-
a
1
/
q
r
∫
a
b
x
n
q
b
-
x
x
-
a
d
x
1
/
q
r
=
1
n
!
b
-
a
1
/
p
-
1
/
q
r
f
n
b
q
r
D
1
r
+
f
n
a
q
r
D
2
r
+
c
b
-
a
1
+
1
/
q
r
×
L
n
q
+
2
n
q
+
2
a
,
b
-
2
A
a
,
b
L
n
q
+
1
n
q
+
1
a
,
b
+
G
2
a
,
b
L
n
q
n
q
a
,
b
1
/
q
r
.This completes the proof of theorem.Remark 18.
The following results are remarkable for Theorem17.
(i) The results obtained in this paper reduce to the results of [24] in case of c=0.
(ii) The results obtained in this paper reduce to the results of [23] in case of r=1.
(iii) The results obtained in this paper reduce to the results of [6] in case of r=1 and c=0.Corollary 19.
Under the conditions Theorem17 for n=1 we have the following inequalities: (40)fbb-faab-a-1b-a∫abfxdx≤b-a1/p-1/qr-1×f′bqD1+f′aqD2+cb-a1+1/qr×Lq+2q+2a,b-2Aa,bLq+1q+1a,b+G2a,bLqqa,b1/q,r≥1f′bqrD1r+f′aqrD2r+cb-a1+1/qr×Lq+2q+2a,b-2Aa,bLq+1q+1a,b+G2a,bLqqa,b1/qr,r≤1Theorem 20.
Forn in N, let f:I⊂0,∞→R be n-times differentiable function on I∘ (interior of I), r>0 and a,b∈I∘ with a<b. If f(n)∈L[a,b] and f(n)q for q>1 is strongly r-convex function with modulus c on [a,b], then the following inequalities hold: (41)Ia,b,n,f≤b-an!Lnpna,br21/rr+1A1/rfnaqr,fnbqr-c6a-b21/q,0<r≤1,1n!b-aLnpna,bLrfnaq,fnbq-c6a-b21/q,r>0,Proof.
For0<r≤1, since f(n)q for q>1 is strongly r-convex function on [a,b], with respect to inequality (6), we have (42)∫abfnxqdx≤b-arr+1fnaqr+fnbqr1/r-c6a-b2.Using Lemma 9 and the Hölder integral inequality we have (43)
I
a
,
b
,
n
,
f
≤
1
n
!
∫
a
b
x
n
f
n
x
d
x
≤
1
n
!
∫
a
b
x
n
p
d
x
1
/
p
∫
a
b
f
n
x
q
d
x
1
/
q
≤
1
n
!
∫
a
b
x
n
p
d
x
1
/
p
b
-
a
r
r
+
1
f
n
a
q
r
+
f
n
b
q
r
1
/
r
-
c
6
a
-
b
2
1
/
q
=
1
n
!
b
n
p
+
1
-
a
n
p
+
1
n
p
+
1
1
/
p
b
-
a
1
/
q
r
r
+
1
f
n
a
q
r
+
f
n
b
q
r
1
/
r
-
c
6
a
-
b
2
1
/
q
=
b
-
a
n
!
b
n
p
+
1
-
a
n
p
+
1
n
p
+
1
b
-
a
1
/
p
r
r
+
1
f
n
a
q
r
+
f
n
b
q
r
1
/
r
-
c
6
a
-
b
2
1
/
q
=
b
-
a
n
!
b
n
p
+
1
-
a
n
p
+
1
n
p
+
1
b
-
a
1
/
p
r
2
1
/
r
r
+
1
f
n
a
q
r
+
f
n
b
q
r
2
1
/
r
-
c
6
a
-
b
2
1
/
q
=
b
-
a
n
!
L
n
p
n
a
,
b
r
2
1
/
r
r
+
1
A
1
/
r
f
n
a
q
r
,
f
n
b
q
r
-
c
6
a
-
b
2
1
/
q
.For r>0, using Lemma 9 and Theorem 2, we have (44)Ia,b,n,f≤1n!∫abxnfnxdx≤1n!∫abxnpdx1/p∫abfnxqdx1/q≤1n!∫abxnpdx1/pb-aLrfnaq,fnbq-c6a-b21/q≤1n!b-aLnpna,bLrfnaq,fnbq-c6a-b21/q.This completes the proof of theorem.Remark 21.
The following results are remarkable for Theorem20.
(i) The results obtained in this paper reduce to the results of [24] in case of c=0.
(ii) The results obtained in this paper reduce to the results of [23] in case of r=1.
(iii) The results obtained in this paper reduce to the results of [6] in case of r=1 and c=0.Corollary 22.
Under the conditions Theorem20 for n=1 we have the following inequalities: (45)fbb-faab-a-1b-a∫abfxdx≤Lpa,br21/rr+1A1/rf′aqr,f′bqr-c6a-b21/q,0<r≤1,Lpa,bLrf′aq,f′bq-c6a-b21/q,r>0,



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*Source: 1219237-2019-01-03.xml*