Research article

Applications of Hölder-İşcan inequality for n-times differentiable (s,m)-convex functions

  • These authors contributed equally to this work and are co-first authors
  • Received: 18 August 2022 Revised: 08 October 2022 Accepted: 10 October 2022 Published: 24 October 2022
  • MSC : 26D07, 26D15, 26E70

  • In this work, Hölder-Isçan inequality is used for the class of n-times differentiable (s,m)-convex functions. The outcomes are new Hermite-Hadamard type inequalities and modified integrals are estimated by better bounds. Special cases are deduced as the existing results from literature. Furthermore, some applications to arithmetic, geometric and logarithmic means are also presented.

    Citation: Khuram Ali Khan, Shaista Ayaz, İmdat İşcan, Nehad Ali Shah, Wajaree Weera. Applications of Hölder-İşcan inequality for n-times differentiable (s,m)-convex functions[J]. AIMS Mathematics, 2023, 8(1): 1620-1635. doi: 10.3934/math.2023082

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  • In this work, Hölder-Isçan inequality is used for the class of n-times differentiable (s,m)-convex functions. The outcomes are new Hermite-Hadamard type inequalities and modified integrals are estimated by better bounds. Special cases are deduced as the existing results from literature. Furthermore, some applications to arithmetic, geometric and logarithmic means are also presented.



    In recent years, convexity theory has gained special attention by many researchers because of it engrossing properties and expedient characterizations. It has many applications in fields like biology, numerical analysis and statistics (see [1,2,3,4]). Mathematical inequalities are extensively studied with all type of convex functions (see[1,3,11,13,14,16]). One of the fundamental inequality is Hermite-Hadamard inequality. It has been discussed via different types of convexities and became the center of attention for many researchers. Recently, in 2016, Khan et al. have discussed generalizations of Hermite-Hadamard type for MT-convex functions [26]. In 2017, Khan et al. studied some new inequalities of Hermite-Hadamard types [27]. In 2019, Khurshid et al. have utilized conformable fractional integrals via preinvex functions [28]. In 2020, Khan et al. have discussed Hermite-Hadamard type inequalities via quantum calculus involving green function [29], Mohammed et al. have established a new version of Hermite-Hadamard inequality for Riemann-Liouville fractional integrals [30], Han et al. used fractional integral to generalize Hermite-Hadamard inequality for convex functions [31], Zhao et al. utilized harmonically convex functions to generalized fractional integral inequalities of Hermite-Hdamrd type [32], Awan et al. presented new inequalities of Hermite-Hdamard type for n-polynomial harmonically convex functions [33]. In 2022, Khan et al. introduced some new versions of Hermite-Hadamard integral inequalities in fuzzy fractional calculus for generalized pre-invex functions via fuzzy-interval-valued settings [34]. This reflects the importance of Hermite Hadamard type inequalities among current research.

    In [9], s-convex function is given as,

    Definition 1.1. A real valued function χ is called s-convex function on R, if

    χ(ςρ+(1ς)γ)ςsχ(ρ)+(1ς)sχ(γ),

    for each ρ,γR and ς(0,1) where s(0,1].

    In [10], m-convexity is discussed as,

    Definition 1.2. A real valued function χ defined on [0,b] is said to be a m-convex function for m[0,1], if

    χ(ςρ+m(1ς)γ)ςχ(ρ)+m(1ς)χ(γ),

    holds for all ρ,γ[0,b] and ς[0,1].

    (s,m)-convexity in [17] is discussed as,

    Definition 1.3. A function χ:[0,b]R, b>0 is said to be a (s,m)-convex function in the second sense where s,m(0,1]2, if

    χ(ςρ+m(1ς)γ)ςsχ(ρ)+m(1ς)sχ(γ),

    holds provided that all ρ,γ[0,b] and ς[0,1].

    Equivalent definition for (s,m)–convex functions:

    Let ρ,α,γ[0,b], ρ<α<γ

    χ(α)(γαγρ)sχ(ρ)+m(αργρ)sχ(γ). (1.1)

    Hölder-İşcan Inequality [5]:

    Let p>1, χ and ψ be real valued functions defined on [ρ,γ] and |χ|p,|ψ|q are integrable functions on interval [ρ,γ]

    γρ|χ(ω)ψ(ω)|dω1γρ(γρ(γω)|χ(ω)|pdω)1p(γρ(γω)|ψ(ω)|qdω)1q+1γρ(γρ(ωρ)|χ(ω)|pdω)1p(γρ(ωρ)|ψ(ω)|qdω)1q, (1.2)

    where 1p+1q=1.

    Following lemma is useful to obtain our main results.

    Lemma 1.4. [8] For nN, let χ:URR be n-times differentiable mapping on U, where ρ,γU, ρ<γ and χnL[ρ,γ], we have following identity

    n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)γρχ(ω)dω=(1)n+1n!γρωnχ(n)(ω)dω, (1.3)

    where an empty set is understood to be nil.

    In this paper, Hölder-İşcan inequality is used to modify inequalities involving functions having s-convex or s-concave derivatives at certain powers. The purpose of this paper is to establish some generalized inequalities for n-times differentiable (s,m)-convex functions. Applications of these inequalities to means are also discussed. Means are defined as,

    Let 0<ρ<γ,

    A(ρ,γ)=ρ+γ2,
    G(ρ,γ)=ργ,
    Lp(ρ,γ)=(γp+1ρp+1(p+1)(γρ))1p,

    where p0,1 and ργ.

    Theorem 2.1. For any positive integer n, let χ:U(0,)R be n-times differentiable mapping on U, where ρ,γU with ρ<γ. If χ(n)L[ρ,γ] and |χ(n)|q for q>1 is (s,m)-convex on interval [ρ,γ] then

    |n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)μρχ(ω)dω|1n!(γρ)1q([γLnpnp(ρ,γ)Lnp+1np+1(ρ,γ)]1p[|χn(γ)|q(s+2)(s+1)+m|χn(ρ)|q(s+2)]1q+[Lnp+1np+1(ρ,γ)ρLnpnp(ρ,γ)]1p[|χn(γ)|q(s+2)+m|χn(ρ)|q(s+1)(s+2)]1q), (2.1)

    where 1p+1q=1.

    Proof. Since |χn|q is (s,m)-convex by using inequality (1.1) for ρ<ω<γ, using Lemma 1.4 and Hölder-Işcan inequality (1.2),

    |χn(ω)|q|χn(ωργργ+mγωγρρ)|q(ωργρ)s|χn(γ)|q+m(γωγρ)s|χn(ρ)|q,|n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)γρχ(ω)dω|1n!γρωn|χ(n)(ω)|dω,1n!1γρ{(γρ(γω)ωnpdω)1p(γρ(γω)|χ(n)(ω)|qdω)1q+(γρ(ωρ)ωnpdω)1p(γρ(ωρ)|χ(n)(ω)|qdω)1q},1n!1γρ(γρ(γω)ωnpdω)1p(γρ(γω)[(ωργρ)s|χn(γ)|q+m(γωγρ)s|χn(ρ)|q]dω)1q+1n!1γρ(γρ(ωρ)ωnpdω)1p(γρ(ωρ)[(ωργρ)s|χn(γ)|q+m(γωγρ)s|χn(ρ)|q]dω)1q, (2.2)

    Let

    I1=[γρ(γω)ωnpdω]1p=[γρ(γωnpωnp+1)dω]1p=(γρ)1p[γ(γnp+1ρnp+1(γρ)(np+1))(γnp+2ρnp+2(γρ)(np+2))]1p=(γρ)1p[γLnpnp(ρ,γ)Lnp+1np+1(ρ,γ)]1p,
    I2=[γρ(ωρ)ωnpdt]1p=[γρ(ωnp+1ρωnp)dω]1p=(γρ)1p[(γnp+2ρnp+2(γρ)(np+2))ρ(γnp+1ρnp+1(γρ)(np+1))]1p=(γρ)1p[Lnp+1np+1(ρ,γ)ρLnpnp(ρ,γ)]1p,
    I3=γρ(γω)(ωρ)sdω=(γω)(ωρ)s+1s+1|γρ+γρ(ωρ)s+1s+1dω=(γρ)s+2(s+1)(s+2),
    I4=γρ(γω)s+1dω=(γρ)s+2s+2,I5=γρ(ωρ)s+1dω=(γρ)s+2s+2,I6=γρ(ωρ)(γω)sdω=(ωρ)(γω)s+1(s+1)|γρ+γρ(γω)s+1(s+1)dω=(γρ)s+2(s+1)(s+2).

    Substituting integrals I1,I2,I3,I4,I5,I6 in inequality (2.2) we have,

    |n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)γρχ(ω)dω|1n!(γρ)((γρ)1p[γLnpnp(ρ,γ)Lnp+1np+1(ρ,γ)]1p[(γρ)2(|χn(γ)|q(s+2)(s+1)+m|χn(ρ)|q(s+2))]1q+(γρ)1p[Lnp+1np+1(ρ,γ)ρLnpnp(ρ,γ)]1p[(γρ)2(|χn(γ)|q(s+2)+m|χn(ρ)|q(s+1)(s+2))]1q)
    =(γρ)1p1+2qn!([γLnpnp(ρ,γ)Lnp+1np+1(ρ,γ)]1p[|χn(γ)|q(s+2)(s+1)+m|χn(ρ)|q(s+2)]1q+[Lnp+1np+1(ρ,γ)ρLnpnp(ρ,γ)]1p[|χn(γ)|q(s+2)+m|χn(ρ)|q(s+1)(s+2)]1q)
    =1n!(γρ)1q([γLnpnp(ρ,γ)Lnp+1np+1(ρ,γ)]1p[|χn(γ)|q(s+2)(s+1)+m|χn(ρ)|q(s+2)]1q+[Lnp+1np+1(ρ,γ)ρLnpnp(ρ,γ)]1p[|χn(γ)|q(s+2)+m|χn(ρ)|q(s+1)(s+2)]1q).

    which is required inequality (2.1).

    For n=1 inequality (2.1) becomes,

    |(χ(γ)γχ(ρ)ργρ)1γργρχ(ω)dω|(γρ)1q1([γLpp(ρ,γ)Lp+1p+1(ρ,γ)]1p[|χ(γ)|q(s+1)(s+2)+m|χ(ρ)|(s+2)q]1q+[Lp+1p+1(ρ,γ)ρLpp(ρ,γ)]1p[m|χ(ρ)|q(s+1)(s+2)+|χ(γ)|(s+2)q]1q). (2.3)

    Remark 2.2. For s=1 and m=1 our resulting inequality (2.1) becomes the inequality (2) of [5].

    Theorem 2.3. For nN, let χ:U(0,)R be n-times differentiable mapping on U, where, ρ,γU, ρ<γ, χ(n)L[ρ,γ] and |χ(n)|q for q>1, is (s,m)-convex on interval [ρ,γ] then following inequality holds

    |n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)γρχ(ω)dω|1s1qn!(12)1p(γρ)2p1((|χ(n)(γ)|q(γρ)s1[Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ)]+m|χ(n)(ρ)|q(γρ)s1[Lnq+2nq+2(ρ,γ)2γLnq+1nq+1(ρ,γ)+γ2Lnqnq(ρ,γ)])1q+(|χ(n)(γ)|q(γρ)s1[Lnq+2nq+2(ρ,γ)2ρLnq+1nq+1(ρ,γ)+ρ2Lnqnq(ρ,γ)]+m|χ(n)(ρ)|q(γρ)s1[Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ)])1q). (2.4)

    Proof. Since |χ(n)|q for q>1 is (s,m)-convex on [ρ,γ], by using Lemma 1.4 and Hölder-İşcan inequality (1.2), since s(0,1], this fact can be used for ω,ρ,γU(0,),

    (ωρ)s<(ωρ)s,(γω)s<(γω)s|n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)γρχ(ω)dω|1n!γρ1.ωn|χ(n)(ω)|dω,1n!1(γρ)([(γρ(γω)dω)1p(γρ(γω)ωnq|χ(n)(ω)|qdω)1q]+[(γρ(ωρ)dω)1p(γρ(ωρ)ωnq|χ(n)(ω)|qdω)1q]),1n!1(γρ)(γρ(γω)dω)1p(γρ(γω)ωnq[(ωργρ)s|χn(γ)|q+m(γωγρ)s|χn(ρ)|q]dt)1q+1n!1(γρ)(γρ(ωρ)dt)1p(γρ(ωρ)ωnq[(ωργρ)s|χn(γ)|q+m(γωγρ)s|χn(ρ)|q]dx)1q,1s1qn!1(γρ)(γρ(γω)dω)1p(γρ(γω)ωnq[(ωρ)(γρ)s|χn(γ)|q+m(γω)(γρ)s|χn(ρ)|q]dω)1q+1s1qn!1(γρ)(γρ(ωρ)dω)1p(γρ(ωρ)ωnq[(ωρ)(γρ)s|χn(γ)|q+m(γω)(γρ)s|χn(ρ)|q]dω)1q,    I1=γρ(γω)dω=(γρ)22    I2=γρ(γω)(ωρ)ωnqdω=γωnq+1nq+1ργωnq+1nq+1ωnq+3nq+3+ρωnq+2nq+2|γρ    =(γnq+3ρnq+3nq+3)+ρ(γnq+2ρnq+2nq+2)+γ(γnq+2ρnq+2nq+2)ργ(γnq+1ρnq+1nq+1)    =(γρ)[Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ)],        I3=γρ(γω)2ωnqdω=γ2ωnq+1nq+1+ωnq+3nq+32γωnq+2nq+2|γρ        =(γnq+3ρnq+3nq+3)2γ(γnq+2ρnq+2nq+2)+γ2(γnq+1ρnq+1nq+1)        =(γρ)[Lnq+2nq+2(ρ,γ)2γLnq+1nq+1(ρ,γ)+γ2Lnqnq(ρ,γ)],        I4=γρ(ωρ)2ωnqdω=ωnq+3nq+3+ρ2ωnq+1nq+12ρωnq+2nq+2|γρ        =(γnq+3ρnq+3nq+3)+ρ2(γnq+1ρnq+1nq+1)2ρ(γnq+2ρnq+2nq+2)        =(γρ)[Lnq+2nq+2(ρ,γ)+ρ2Lnqnq(ρ,γ)2ρLnq+1nq+1(ρ,γ)]. (2.5)

    Substituting integrals I1,I2,I3,I4,I5,I6 in inequality (2.5) we have,

    |n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)γρχ(ω)dω|1s1qn!(12)1p(γρ)2p1×((|χ(n)(γ)|q(γρ)s[(γρ)(Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ))]+m|χ(n)(ρ)|q(γρ)s[(γρ)(Lnq+2nq+2(ρ,γ)2γLnq+1nq+1(ρ,γ)+γ2Lnqnq(ρ,γ))])1q+(|χ(n)(γ)|q(γρ)s[(γρ)(Lnq+2nq+2(ρ,γ)2ρLnq+1nq+1(ρ,γ)+ρ2Lnqnq(ρ,γ))]+m|χ(n)(ρ)|q(γρ)s[(γρ)(Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ))])1q),
    =1s1qn!(12)1p(γρ)2p1×((|χ(n)(γ)|q(γρ)s1[Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ)]+m|χ(n)(ρ)|q(γρ)s1[Lnq+2nq+2(ρ,γ)2γLnq+1nq+1(ρ,γ)+γ2Lnqnq(ρ,γ)])1q+(|χ(n)(γ)|q(γρ)s1[Lnq+2nq+2(ρ,γ)2ρLnq+1nq+1(ρ,γ)+ρ2Lnqnq(ρ,γ)]+m|χ(n)(ρ)|q(γρ)s1[Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ)])1q).

    For n=1, Theorem2.3 reduced to the inequality

    |γχ(γ)ρχ(ρ)(γρ)1(γρ)γρχ(ω)dω|1s1q(12)1p(γρ)2p2((|χ(1)(γ)|q(γρ)s1[Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)ργLqq(ρ,γ)]+m|χ(1)(ρ)|(γρ)s1q[Lq+2q+2(ρ,γ)2γLq+1q+1(ρ,γ)+γ2Lqq(ρ,γ)])1q+(|χ(1)(γ)|(γρ)s1q[Lq+2q+2(ρ,γ)2ρLq+1q+1(ρ,γ)+ρ2Lqq(ρ,γ)]+m|χ(1)(ρ)|(γρ)s1q[Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)ργLqq(ρ,γ)])1q). (2.6)

    Remark 2.4. For s=1 and m=1 our resulting inequality (2.4) becomes the inequality (6) of [5].

    Theorem 2.5. If function χ:[0,b]R, b>0 is a (s, m)-convex function in the second sense where (s,m)(0,1]2, holds provided that all ρ,γ[0,b] and ς[0,1], then

    2sχ(ρ+mγ2)[1mγρmγρχ(ω)dω+m2mγργρmχ(l)dl]χ(ρ)+mχ(γ)s+1+χ(γ)+mχ(ρm2)s+1. (2.7)

    Proof. A function χ:[0,b]R, b>0 is said to be a (s,m)-convex function in the second sense where s,m(0,1]2, if

    χ(ςρ+m(1ς)γ)ςsχ(ρ)+m(1ς)sχ(γ),

    holds provided that all ρ,γ[0,b] and ς[0,1].

    Integrating w.r.t ς on [0,1],

    10χ(ςρ+m(1ς)γ)dς10ςsχ(ρ)dς+10m(1ς)sχ(γ)dς,=ςs+1s+1|10χ(ρ)mχ(γ)(1ς)s+1s+1|10=χ(ρ)+mχ(γ)s+1.      10χ(ςρ+m(1ς)γ)dςχ(ρ)+mχ(γ)s+1. (2.8)

    and

    χ(ςγ+m(1ς)ρm2)ςsχ(γ)+m(1ς)sχ(ρm2),10χ(ςγ+m(1ς)ρm2)dςχ(γ)+mχ(ρm2)s+1. (2.9)

    As χ is (s,m)-convex,

    χ(ρ+mγ2)=χ(ςρ+(1ς)mγ2+m.(1ς)ρm+ςγ2)(12)sχ(ςρ+(1ς)γm)+m(12)sχ(ςγ+(1ς)ρm),

    Integrating w.r.t ς over [0,1] and by using (2.8) and (2.9) we get,

    2sχ(ρ+mγ2)10(χ(ςρ+(1ς)γm)dς+m10χ(ςγ+(1ς)ρm)dςχ(ρ)+mχ(γ)s+1+χ(γ)+mχ(ρm2)s+1. (2.10)

    Substituting in first integral,

    ςρ+(1ς)γm=ω,

    10χ(ςρ+(1ς)mγ)dς=1γmργmρχ(ω)dω. (2.11)

    Substituting in the second integral,

    ςγ+(1ς)ρm=l,

    10χ(ςγ+(1ς)ρm)dς=mγmργρmχ(l)dl, (2.12)

    Using (2.11) and (2.12) in (2.10) required inequality (2.7) obtained.

    Remark 2.6. For s,m=1 inequality (2.7) becomes classical Hadamard inequality for convex functions.

    Theorem 2.7. For nN, let χ:U(0,)R be n-times differentiable mapping on U, where, ρ,γU, ρ<γ and χ(n)L[ρ,γ] and |χ(n)|q for q>1 is (s, m)-concave on interval [ρ,mγ], then

    |n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)mγρχ(ω)dω|2sq(mγρ)1q|χ(n)(ρ+mγ2)|n!((γLnpnp(ρ,mγ)Lnp+1np+1(ρ,mγ))1p+(Lnp+1np+1(ρ,mγ)ρLnpnp(ρ,mγ))1p). (2.13)

    Proof. |χ(n)|q for q>1 is (s,m)-concave then by using Theorem 2.5 we have,

    |χ(n)(ρ)|q+m|χ(n)(γ)|qs+1+|χ(n)(γ)|q+m|χ(n)(ρm2)|qs+1m2(mγρ)γρm|χ(n)(l)|qdl1(mγρ)mγρ|χ(n)(ω)|qdω2s|χ(n)(ρ+mγ2)|q,
    mγρ|χ(n)(ω)|qdω2s(mγρ)|χ(n)(ρ+mγ2)|q,
    1(mγρ)γmρ(γω)|χ(n)(ω)|qdωγmρ|χ(n)(ω)|qdω2s(mγρ)|χ(n)(ρ+mγ2)|q,
    1(mγρ)γmρ(γω)|χ(n)(ω)|qdωγmρ|χ(n)(ω)|qdω2s(mγρ)|χ(n)(ρ+mγ2)|q.

    Using Lemma 1.4 and Hölder-Îşcan inequality (1.2),

    |n1ν=0(1)ν(χ(ν)(γ)γν+1χ(ν)(ρ)ρν+1(ν+1)!)γmρχ(ω)dω|1n!γmρωn|χ(n)(ω)|dω,      1n!1γρ{(γmρ(γω)ωnpdω)1p(mγρ(γω)|χn(ω)|qdω)1q+(γmρ(ωρ)ωnpdω)1p(mγρ(ωρ)|χn(ω)|qdω)1q},      1n!1γρ((γmρ(γω)ωnpdω)1p(2s(mγρ)2|χ(n)(ρ+mγ2)|q)1q+(γmρ(ωρ)ωnpdω)1p(2s(mγρ)2|χ(n)(ρ+mγ2)|q)1q),            I1=(γmρ(γω)ωnpdω)1p=(γωnp+1np+1|γmρωnp+2np+2|γmρ)1p            =(mγρ)1p(γLnpnp(ρ,mγ)Lnp+1np+1(ρ,mγ))1p,            I2=(γmρ(ωρ)ωnpdω)1p=(ωnp+2np+2|γmρρωnp+1np+1|γmρ)1p            =(mγρ)1p(Lnp+1np+1(ρ,mγ)ρLnpnp(ρ,mγ))1p. (2.14)

    Substituting integrals I1,I2 in inequality (2.14) required inequality (2.13) is obtained.

    For n=1 inequality (2.13) becomes,

    |χ(γ)γρχ(ρ)(γρ)1(γρ)γmρχ(ω)dω|2sq(mγρ)1q|χ(1)(ρ+γ2)|1!((γLpp(ρ,mγ)Lp+1p+1(ρ,mγ))1p+(Lp+1p+1(ρ,mγ)ρLpp(ρ,mγ))1p). (2.15)

    Remark 2.8. For s=1 and m=1 our resulting inequality becomes the inequality obtained in Theorem 4 of [5].

    Proposition 2.9. Let ρ,γ(0,), where ρ<γ, q>1, n,iN with in,

    |Lii(ρ,γ)[(i+1)n1ν=0(1)νP(i,ν)(ν+1)!1]|1n!(γρ)1q1×([γLnpnp(ρ,γ)Lnp+1np+1(ρ,γ)]1p(γ(in)q(s+1)(s+2)+mρ(in)q(s+2))1q+[Lnp+1np+1(ρ,γ)ρLnpnp(ρ,γ)]1p(mρ(in)q(s+1)(s+2)+γ(in)q(s+2))1q), (2.16)

    where

    P(i,n)={i(i1)...(in+1),i>nn!,i=n1,n=0}.

    Proof. Let

    χ(ω)=ωi,|χ(n)(ω)|q=|P(i,n)ωin|q

    Let

    g(ς)=|P(i,n)(ςρ+m(1ς)γ|(in)q|P(i,n)ςsρ|(in)q|mP(i,n)(1ς)sγ|(in)q,
    g(ς)=P(i,n)((in)q)((in)q1)(ςρ+m(1ς)γ)(in)q2(ρmγ)2s(s1)ςs2P(i,n)ρ(in)qms(s1)(1ς)s2P(i,n)γ(in)q,

    g(ς)0 means g is convex and g(1)=g(0)=0, which omplies g0, hence

    |P(i,n)(ςρ+m(1ς)γ)|(in)q|P(i,n)ςsρ|(in)q+|mP(i,n)(1ς)sγ)|(in)q.

    By using Theorem 2.1 for |χn(ω)|q which is (s,m)–convex for s,m(0,1]2 inequality (2.16) obtained.

    Remark 2.10. For s,m=1 inequality (2.16) becomes inequality (3) of [5].

    Example 2.11. Taking i=2, n=1, p=q=2 in Proposition 2.9, the following is valid:

    2A(ρ2,γ2)+G2(ρ,γ)(326)([A(3ρ2,γ2)+G2(ρ,γ)]12(γ2(s+1)(s+2)+mρ2(s+2))12+[A(ρ2,3γ2)+G2(ρ,γ)]12(mρ2(s+1)(s+2)+γ2(s+2))12),

    where A and G are classical arithmetic and geometric means, respectively.

    Proposition 2.12. Let ρ,γ(0,), with, ρ<γ, q>1 and nN,

    1(γρ)1q1([γLpp(ρ,γ)Lp+1p+1(ρ,γ)]1p[(γq(s+1)(s+2)+mρq(s+2))]1q+[Lp+1p+1(ρ,γ)ρLpp(ρ,γ)]1p[(mρq(s+1)(s+2)+γq(s+2))]1q), (2.17)

    where L is classical logarithmic mean.

    Proof.

    χ(ω)=lnω,|χ(1)(ω)|q=|ω1|q

    Let

    g(ς)=|(ςρ+m(1ς)γ|q|ςsρ|q|m(1ς)sγ|q
    g(ς)=(q)(q1)(ςρ+m(1ς)γ)q2(ρmγ)2s(s1)ςs2ρqms(s1)(1ς)s2γq,

    g(ς)0 means g is convex and g(1)=g(0)=0 which implies g0 as

    |(ςρ+m(1ς)γ|q|ςsρ|q+|m(1ς)sγ|q.

    So |χ(1)(ω)|q is (s,m)-convex. Then by using inequality (2.3) required inequality (2.17) obtained.

    Remark 2.13. For s,m=1 inequality (2.17) becomes (4) of [5].

    Example 2.14. For n=1 and p=q=2, Proposition 2.12 gives:

    116([A(3ρ2,γ2)+G2(ρ,γ)]1p[(γ2(s+1)(s+2)+mρ2(s+2))]12+[A(ρ2,3γ2)+G2(ρ,γ)]1p[(mρ2(s+1)(s+2)+γ2(s+2))]12).

    Proposition 2.15. Let ρ,γ(0,), ρ<γ, q>1, i(,0][1,){2q,q}

    then

    Liq+1iq+1(ρ,γ)(γρ)1q1([γLpp(ρ,γ)Lp+1p+1(ρ,γ)]1p[(γi(s+1)(s+2)+mρi(s+2))]1q+[Lp+1p+1(ρ,γ)ρLpp(ρ,γ)]1p[(mρi(s+1)(s+2)+γi(s+2))]1q). (2.18)

    Proof.

    χ(t)=qi+qωiq+1,|χ(ω)|q=ωi

    Let

    g(ς)=|(ςρ+m(1ς)γ|i|ςsρ|i|m(1ς)sγ|i,
    g(ς)=(i)(i1)(ςρ+m(1ς)γ)i2(ρmγ)2s(s1)ςs2ρims(s1)(1ς)s2γi,

    g(ς)0 and g(1)=g(0) so g0 and |χ(ω)|q is (s,m)-convex, by using inequality (2.3) we have (2.18).

    Remark 2.16. For s,m=1 inequality (2.18) becomes (5) of [5].

    Example 2.17. For i=2 and p=q=2 Proposition 2.15 reduced to

    2A(ρ2,γ2)+G2(ρ,γ)(36)([A(3ρ2,γ2)+G2(ρ,γ)]12[(γ2(s+1)(s+2)+mρ2(s+2))]12+[A(ρ2,3γ2)+G2(ρ,γ)]12[(mρ2(s+1)(s+2)+γ2(s+2))]12). (2.19)

    Proposition 2.18. Let ρ,γ(0,) with ρ<γ, q>1 and nN then we have

    ×|Lii(ρ,γ)[n1ν=0(1)νP(i,ν)(ν+1)!1]|P(i,n)s1qn!(12)1p(γρ)2p1(γ(in)q(γρ)s1[Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ)]+mρ(in)q(γρ)s1[Lnq+2nq+2(ρ,γ)2γLnq+1nq+1(κ,μ)+μ2Lnqnq(ρ,γ)])1q+P(i,n)s1qn!(12)1p(γρ)2p1(γ(in)q(γρ)s1[Lnq+2nq+2(ρ,γ)2ρLnq+1nq+1(ρ,γ)+ρ2Lnqnq(ρ,γ)]+mρ(in)q(γρ)s1[Lnq+2nq+2(ρ,γ)+(ρ+γ)Lnq+1nq+1(ρ,γ)ργLnqnq(ρ,γ)])1q. (2.20)

    Proof. Let,

    χ(ω)=ωi,|χ(n)(ω)|q=[P(i,n)ωin]q

    As |χn(ω)|q is (s,m)-convex on (0,), therefore by using Theorem 2.3 required inequality (2.20) is obtained.

    Remark 2.19. For s,m=1 inequality (2.20) becomes inequality obtained in Proposition 4 of [5].

    Proposition 2.20. Let ρ,γ(0,) with ρ<γ q>1 and nN then we have,

    1(γρ)2p2s1q.21p((γq(γρ)s1[Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)ργLqq(ρ,γ)]+mρq(γρ)s1[Lq+2q+2(ρ,γ)2γLq+1q+1(ρ,γ)+γ2Lqq(ρ,γ)])1q+(γq(γρ)s1[Lq+2q+2(ρ,γ)2ρLq+1q+1(ρ,γ)+ρ2Lqq(ρ,γ)]+mρq(γρ)s1[Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)ργLqq(ρ,γ)])1q), (2.21)

    Proof.

    χ(ω)=lnω,|χ(1)(ω)|q=[ω1]q

    As |χ(1)(ω)|q is (s,m)–convex, therefore by using inequality (2.6) required (2.21) obtained.

    Remark 2.21. For s,m=1 inequality (2.21) becomes inequality obtained in Proposition 5 of [5].

    Proposition 2.22. Let ρ,γ(0,) with ρ<γ q>1 and i(,0]{2q,q}, then

    Liq+1iq+1(ρ,γ)(γρ)2p2s1q.21p((γi(γρ)s1[Lnq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)ργLqq(ρ,γ)]+mρi(γρ)s1[Lq+2q+2(ρ,γ)2γLq+1q+1(ρ,γ)+γ2Lqq(ρ,γ)])1q+(γi(γρ)s1[Lq+2q+2(ρ,γ)2ρLq+1q+1(ρ,γ)+ρ2Lqq(ρ,γ)]+mρm(γρ)s1[Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)ργLqq(ρ,γ)])1q). (2.22)

    Proof.

    χ(ω)=qi+qωiq+1|χ(ω)|q=ωi

    |χ(w)|q is (s,m)-convex by using inequality (2.6) required (2.22) obtained.

    For i=1 inequality (2.22) becomes,

    L1q+11q+1(ρ,γ)(γρ)2p2s1q.21p((γ1(γρ)s1[Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)ργLqq(ρ,γ)]+mρ1(γρ)s1[Lq+2q+2(ρ,γ)2γLq+1q+1(ρ,γ)+γ2Lqq(ρ,γ)])1q+(γ1(γρ)s1[Lq+2q+2(ρ,γ)2ρLq+1q+1(ρ,γ)+ρ2Lqq(ρ,γ)]+mρ1(γρ)s1[Lq+2q+2(ρ,γ)+(ρ+γ)Lq+1q+1(ρ,γ)ργLqq(ρ,γ)])1q). (2.23)

    Remark 2.23. For s,m=1 inequality (2.22) becomes inequality obtained in Proposition 6 of [5].

    Proposition 2.24. Let ρ,γ(0,) with ρ<γ, q>1 and i[0,1] we have,

    Liq+1iq+1(ρ,γ)2sq(mγρ)1q1!Aiq(ρ,γ)((γLpp(ρ,mγ)Lp+1p+1(ρ,mγ))1p+(Lp+1p+1(ρ,mγ)ρLpp(ρ,mγ))1p). (2.24)

    Proof.

    χ(ω)=qi+qωiq+1,|χ(ω)|q=ωi.

    As |χ(ω)|q is (s,m)-concave by using inequality (2.15) we obtain required inequality (2.24).

    Remark 2.25. For s,m=1 inequality (2.24) becomes the inequality obtained in Proposition 9 of [5].

    In this paper, Hölder-Isçan inequality is utilized to prove Hermite-Hadamard type inequalities for n-times differentiable (s,m)-convex functions. The method is adequate and provide many generalizations of existing results as shown in remarks. Moreover, many other inequalities can be generalized for other types of convex functions.

    This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation, (grant number B05F650018)

    The authors declare no conflict of interest.



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