Research article

New weighted generalizations for differentiable exponentially convex mapping with application

  • Received: 19 December 2019 Accepted: 25 March 2020 Published: 10 April 2020
  • MSC : 26D10, 26D15, 26E60

  • The main aim of the present paper is to present a novel approach base on the exponentially convex function to broaden the utilization of celebrated Hermite-Hadamard type inequality. The proposed technique presents an auxiliary result of constructing the set of base functions and gives deformation equations in a simple form. The auxiliary result in the convexity has provided a convenient way of establishing the convergence region of several novel results. The strategy is not limited to the small parameter, such as in the classical method. The numerical examples obtained by the proposed approach indicate that the approach is easy to implement and computationally very attractive. The implementation of this numerical scheme clearly exhibits its effectiveness, reliability, and easiness regarding the applications in error estimates for weighted mean, the integral formula, rth moments of a continuous random variable, application to weighted special means and in developing the variants by extraordinary choices of n and θ as well as its better approximation.

    Citation: Saima Rashid, Rehana Ashraf, Muhammad Aslam Noor, Khalida Inayat Noor, Yu-Ming Chu. New weighted generalizations for differentiable exponentially convex mapping with application[J]. AIMS Mathematics, 2020, 5(4): 3525-3546. doi: 10.3934/math.2020229

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  • The main aim of the present paper is to present a novel approach base on the exponentially convex function to broaden the utilization of celebrated Hermite-Hadamard type inequality. The proposed technique presents an auxiliary result of constructing the set of base functions and gives deformation equations in a simple form. The auxiliary result in the convexity has provided a convenient way of establishing the convergence region of several novel results. The strategy is not limited to the small parameter, such as in the classical method. The numerical examples obtained by the proposed approach indicate that the approach is easy to implement and computationally very attractive. The implementation of this numerical scheme clearly exhibits its effectiveness, reliability, and easiness regarding the applications in error estimates for weighted mean, the integral formula, rth moments of a continuous random variable, application to weighted special means and in developing the variants by extraordinary choices of n and θ as well as its better approximation.


    The classical convexity and concavity of functions are two fundamental notions in mathematics, they have widely applications in many branches of mathematics and physics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. The origin theory of convex functions is generally attributed to Jensen [31]. The well-known book [32] played an indispensable role in the the theory of convex functions.

    The significance of inequalities is increasing day by day in the real world because of their fertile applications in our life and used to solve many complex problems in all areas of science and technology [33,34,35,36,37,38,39,40]. Integral inequalities have numerous applications in number theory, combinatorics, orthogonal polynomials, hypergeometric functions, quantum theory, linear programming, optimization theory, mechanics and in the theory of relativity [41,42,43,44,45,46,47,48]. This subject has received considerable attention from researchers [49,50,51,52,53,54] and hence it is assumed as an incorporative subject between mathematics, statistics, economics, and physics [55,56,57,58,59,60].

    One of the most well known and considerably used inequalities for convex function is the Hermite-Hadamard inequality, which can be stated as follows.

    Let IR be an interval, Y:IR be a convex function. Then the double inequality

    Y(ρ1+ρ22)1ρ2ρ1ρ2ρ1Y(ϱ)dϱY(ρ1)+Y(ρ2)2 (1.1)

    holds for all ρ1,ρ2I with ρ1ρ2. If Y is concave on the interval I, then the reversed inequality (1.1) holds.

    The Hermite-Hadamard inequality (1.1) has wide applications in the study of functional analysis (geometry of Banach spaces) and in the field of non-linear analysis [61]. Interestingly, both sides of the above integral inequality (1.1) can characterize the convex functions.

    Closely related to the convex (concave) functions, we have the concept of exponentially convex (concave) functions. The exponentially convex (concave) functions can be considered as a noteworthy extension of the convex functions and have potential applications in information theory, big data analysis, machine learning, and statistics [62,63]. Bernstein [64] and Antczak [65] introduced these exponentially convex functions implicitly and discuss their role in mathematical programming. Dragomir and Gomm [66] and Rashid et al. [67] established novel outcomes for these exponentially convex functions.

    Now we recall the concept of exponentially convex functions, which is mainly due to Awan et al. [68].

    Definition 1.1. ([68]) Let θR. Then a real-valued function Y:[0,)R is said to be θ-exponentially convex if

    Y(τρ1+(1τ)ρ2)τeθρ1Y(ρ1)+(1τ)eθρ2Y(ρ2) (1.2)

    for all ρ1,ρ2[0,) and τ[0,1]. Inequality (1.2) will hold in the reverse direction if Y is concave.

    For example, the mapping Y:RR, defined by Y(υ)=υ2 is a concave function, thus this mapping is an exponentially convex for all θ>0. Exponentially convex functions are employed for statistical analysis, recurrent neural networks, and experimental designs. The exponentially convex functions are highly useful due to their dominant features.

    Recall the concept of exponentially quasi-convex function, introduced by Nie et al. [69].

    Definition 1.2. ([69]) Let θR. Then a mapping Y:[0,)RR is said to be θ-exponentially quasi-convex if

    Y(τρ1+(1τ)ρ2)max{eθρ1Y(ρ1),eθρ2Y(ρ2)}

    for all ρ1,ρ2[0,) and τ[0,1].

    Kirmaci [70], and Pearce and Pečarič [71] established the new inequalities involving the convex functions as follows.

    Theorem 1.3. ([70]) Let IR be an interval, ρ1,ρ1I with ρ1<ρ2, and Y:IR be a differentiable mapping on I (where and in what follows I denotes the interior of I) such that YL([ρ1,ρ2]) and |Y| is convex on [ρ1,ρ2]. Then

    |Y(ρ1+ρ22)1ρ2ρ1ρ2ρ1Y(ϱ)dϱ|(ρ2ρ1)(|Y(ρ1)|+|Y(ρ2)|)8. (1.3)

    Theorem 1.4. ([71]) Let λR with λ0, IR be an interval, ρ1,ρ1I with ρ1<ρ2, and Y:IR be a differentiable mapping on I such that YL([ρ1,ρ2]) and |Y|λ is convex on [ρ1,ρ2]. Then

    |Y(ρ1+ρ22)1ρ2ρ1ρ2ρ1Y(ϱ)dϱ|(ρ2ρ1)4[|Y(ρ1)|λ+|Y(ρ2)|2]1λ. (1.4)

    The principal objective of this work is to determine the novel generalizations for weighted variants of (1.3) and (1.4) associated with the class of functions whose derivatives in absolute value at certain powers are exponentially convex with the aid of the auxiliary result. Moreover, an analogous improvement is developed for exponentially quasi-convex functions. Utilizing the obtained consequences, some new bounds for the weighted mean formula, rth moments of a continuous random variable and special bivariate means are established. The repercussions of the Hermite-Hadamard inequalities have depicted the presentations for various existing outcomes. Results obtained by the application of the technique disclose that the suggested scheme is very accurate, flexible, effective and simple to use.

    In what follows we use the notations

    L(ρ1,ρ2,τ)=n+τn+1ρ1+1τn+1ρ2

    and

    M(ρ1,ρ2,τ)=1τn+1ρ1+n+τn+1ρ2

    for τ[0,1] and all nN.

    From now onwards, let ρ1,ρ2R with ρ1<ρ2 and I=[ρ1,ρ2], unless otherwise specified. The following lemma presented as an auxiliary result which will be helpful for deriving several new results.

    Lemma 2.1. Let nN, Y:IR be a differentiable mapping on I such that YL1([ρ1,ρ2]), and U:[ρ1,ρ2][0,) be differentiable mapping. Then one has

    12[U(ρ1)[Y(ρ1)+Y(ρ2)]{U(nρ1+ρ2n+1)U(ρ1+nρ2n+1)+U(ρ2)}Y(nρ1+ρ2n+1)
    {U(nρ1+ρ2n+1)U(ρ1+nρ2n+1)+U(ρ2)}Y(nρ1+ρ2n+1)]+ρ2ρ12(n+1)10{[Y(n+τn+1ρ1
    1τn+1ρ2)+Y(1τn+1ρ1+n+τn+1ρ2)][U(n+τn+1ρ1+1τn+1ρ2)+U(1τn+1ρ1+n+τn+1ρ2)]}dτ
    =ρ2ρ12(n+1){10[U(n+τn+1ρ1+1τn+1ρ2)U(1τn+1ρ1+n+τn+1ρ2)+U(ρ2)]
    ×[Y(n+τn+1ρ1+1τn+1ρ2)+Y(1τn+1ρ1+n+τn+1ρ2)]dτ}. (2.1)

    Proof. It follows from integration by parts that

    I1=10[U(n+τn+1ρ1+1τn+1ρ2)U(1τn+1ρ1+n+τn+1ρ2)+U(ρ2)]Y(n+τn+1ρ1+1τn+1ρ2)dτ
    =n+1ρ2ρ1{U(n+τn+1ρ1+1τn+1ρ2)U(1τn+1ρ1+n+τn+1ρ2)+U(ρ2)}Y(n+τn+1ρ1+1τn+1ρ2)|10
    ρ1ρ2n+110Y(n+τn+1ρ1+1τn+1ρ2)[U(n+τn+1ρ1+1τn+1ρ2)+U(1τn+1ρ1+n+τn+1ρ2)]dτ
    =n+1ρ2ρ1[U(ρ1)Y(ρ1)[U(nρ1+ρ2n+1)U(ρ1+nρ2n+1)+U(ρ2)]]Y(nρ1+ρ2n+1)
    +10Y(n+τn+1ρ1+1τn+1ρ2)[U(n+τn+1ρ1+1τn+1ρ2)+U(1τn+1ρ1+n+τn+1ρ2)]dτ.

    Similarly, we have

    I2=10[U(n+τn+1ρ1+1τn+1ρ2)U(1τn+1ρ1+n+τn+1ρ2)+U(ρ2)]Y(1τn+1ρ1+n+τn+1ρ2)dτ
    =n+1ρ2ρ1[U(ρ1)Y(ρ1)[U(nρ1+ρ2n+1)U(ρ1+nρ2n+1)+U(ρ2)]]Y(nρ1+ρ2n+1)
    +10Y(1τn+1ρ1+n+τn+1ρ2)[U(n+τn+1ρ1+1τn+1ρ2)+U(1τn+1ρ1+n+τn+1ρ2)]dτ.

    Adding I1 and I2, then multiplying by ρ2ρ12(n+1) we get the desired identity (2.1).

    Theorem 2.2. Let nN, θR, Y:IR be a differentiable mapping on I such that |Y| is θ-exponentially convex on I, and V:I[0,) be a continuous and positive mapping such it is symmetric with respect to nρ1+ρ2n+1. Then

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    ρ2ρ1n+1[|eθρ1Y(ρ1)|+|eθρ2Y(ρ2)|]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ. (2.2)

    Proof. Let τ[ρ1,ρ2] and Y(τ)=τρ1V(ϱ)dϱ. Then it follows from Lemma 2.1 that

    ρ2ρ12(n+1)10[Y(n+τn+1ρ1+1τn+1ρ2)+Y(1τn+1ρ1+n+τn+1ρ2)][V(n+τn+1ρ1+1τn+1ρ2)
    +V(1τn+1ρ1+n+τn+1ρ2)]dτY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ
    =ρ2ρ12(n+1)10{L(ρ1,ρ2,τ)ρ1V(ϱ)dϱ+ρ2M(ρ1,ρ2,τ)V(ϱ)dϱ}
    ×[Y(n+τn+1ρ1+1τn+1ρ2)+Y(1τn+1ρ1+n+τn+1ρ2)]dτ. (2.3)

    Since V(ϱ) is symmetric with respect to ϱ=nρ1+ρ2n+1, we have

    ρ2ρ12(n+1)10[Y(n+τn+1ρ1+1τn+1ρ2)+Y(1τn+1ρ1+n+τn+1ρ2)][V(n+τn+1ρ1+1τn+1ρ2)
    +V(1τn+1ρ1+n+τn+1ρ2)]dτ
    =ρ2ρ1(n+1)10Y(n+τn+1ρ1+1τn+1ρ2)V(n+τn+1ρ1+1τn+1ρ2)dτ
    +ρ2ρ1(n+1)10Y(1τn+1ρ1+n+τn+1ρ2)V(1τn+1ρ1+n+τn+1ρ2)dτ
    =nρ1+ρ2n+1ρ1Y(ϱ)V(ϱ)dϱ+ρ2ρ1+nρ2n+1Y(ϱ)V(ϱ)dϱ=ρ2ρ1Y(ϱ)V(ϱ)dϱ (2.4)

    and

    L(ρ1,ρ2,τ)ρ1V(ϱ)dϱ=ρ2M(ρ1,ρ2,τ)V(ϱ)dϱτ[0,1]. (2.5)

    From (2.3)–(2.5) we clearly see that

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    ρ2ρ1n+1{10L(ρ1,ρ2,τ)ρ1|Y(n+τn+1ρ1+1τn+1ρ2)|dτ+10L(ρ1,ρ2,τ)ρ1|Y(1τn+1ρ1+n+τn+1ρ2)|dτ}. (2.6)

    Making use of the exponentially convexity of |Y| we get

    10L(ρ1,ρ2,τ)ρ1V(ϱ)|Y(n+τn+1ρ1+1τn+1ρ2)|dϱdτ+10L(ρ1,ρ2,τ)ρ1V(ϱ)|Y(1τn+1ρ1+n+τn+1ρ2)|dϱdτ
    10L(ρ1,ρ2,τ)ρ1V(ϱ)[n+τn+1|eθρ1Y(ρ1)|+1τn+1|eθρ2Y(ρ2)|+1τn+1|eθρ1Y(ρ1)+n+τn+1|eθρ2Y(ρ2)||]dϱdτ
    =[|eθρ1Y(ρ1)|+|eθρ2Y(ρ2)|]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ. (2.7)

    Therefore, inequality (2.2) follows from (2.6) and (2.7).

    Corollary 2.1. Let θ=0. Then Theorem 2.2 leads to

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    ρ2ρ1n+1[|Y(ρ1)|+|Y(ρ2)|]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    Corollary 2.2. Let n=1. Then Theorem 2.2 reduces to

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(ρ1+ρ22)ρ2ρ1V(ϱ)dϱ|
    ρ2ρ12[|eθρ1Y(ρ1)|+|eθρ2Y(ρ2)|]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    Corollary 2.3. Let V(ϱ)=1. Then then Theorem 2.3 becomes

    |Y(nρ1+ρ2n+1)1ρ2ρ1ρ2ρ1Y(ϱ)dϱ|
    ρ2ρ12(n+1)2[|eθρ1Y(ρ1)|+|eθρ2Y(ρ2)|].

    Remark 2.1. Theorem 2.2 leads to the conclusion that

    (1) If n=1 and θ=0, then we get Theorem 2.2 of [72].

    (2) If n=V(ϱ)=1 and θ=0, then we obtain inequality (1.2) of [70]

    Theorem 2.3. Taking into consideration the hypothesis of Theorem 2.2 and λ1. If θR and |Y|λ is θ-exponentially convex on I, then

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    2(ρ2ρ1)n+1[|eθρ1Y(ρ1)|λ+|eθρ2Y(ρ2)|λ2]1λ10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ (2.8)

    for all nN.

    Proof. Continuing inequality (2.6) in the proofs of Theorem 2.2 and using the well-known Hölder integral inequality, one has

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    ρ2ρ1n+1{(10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ)11λ(10L(ρ1,ρ2,τ)ρ1V(ϱ)|Y(n+τn+1ρ1+1τn+1ρ2)|λdϱdτ)1λ
    +(10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ)11λ(10L(ρ1,ρ2,τ)ρ1V(ϱ)|Y(1τn+1ρ1+n+τn+1ρ2)|λdϱdτ)1λ}
    ρ2ρ1n+1(10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ)11λ{(10L(ρ1,ρ2,τ)ρ1V(ϱ)|Y(n+τn+1ρ1+1τn+1ρ2)|λdϱdτ)1λ
    +(10L(ρ1,ρ2,τ)ρ1V(ϱ)|Y(1τn+1ρ1+n+τn+1ρ2)|λdϱdτ)1λ}. (2.9)

    It follows from the power-mean inequality

    μa+νa<21a(μ+ν)a

    for μ,ν>0 and a<1 that

    (10L(ρ1,ρ2,τ)ρ1V(ϱ)|Y(n+τn+1ρ1+1τn+1ρ2)|λdϱdτ)1λ (2.10)
    +(10L(ρ1,ρ2,τ)ρ1V(ϱ)|Y(1τn+1ρ1+n+τn+1ρ2)|λdϱdτ)1λ
    211λ{10L(ρ1,ρ2,τ)ρ1V(ϱ)(|Y(n+τn+1ρ1+1τn+1ρ2)|λ+|Y(1τn+1ρ1+n+τn+1ρ2)|λ)dϱdτ}1λ.

    Since |Y|λ is an θ-exponentially convex on I, we have

    |Y(n+τn+1ρ1+1τn+1ρ2)|λ+|Y(1τn+1ρ1+n+τn+1ρ2)|
    n+τn+1|eθρ1Y(ρ1)|q+1τn+1|eθρ2Y(ρ2)|q+1τn+1|eθρ1Y(ρ1)|q+n+τn+1|eθρ2Y(ρ2)|q
    =|eθρ1Y(ρ1)|q+|eθρ2Y(ρ2)|q. (2.11)

    Combining (2.9)–(2.11) gives the required inequality (2.8).

    Corollary 2.4. Let n=1. Then Theorem 2.3 reduces to

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(ρ1+ρ22)ρ2ρ1V(ϱ)dϱ|
    (ρ2ρ1)[|eθρ1Y(ρ1)|λ+|eθρ2Y(ρ2)|λ2]1λ10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    Corollary 2.5. Let θ=0. Then Theorem 2.3 leads to

    |ρ2ρ1Y(x)V(x)dxY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    2(ρ2ρ1)n+1[|Y(ρ1)|λ+|Y(ρ2)|λ2]1λ10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    Corollary 2.6. Let V(ϱ)=1. Then Theorem 2.3 becomes

    |Y(nρ1+ρ2n+1)1ρ2ρ1ρ2ρ1Y(ϱ)dϱ|(ρ2ρ1)2(n+1)[|Y(ρ1)|λ+|Y(ρ2)|λ2]1λ.

    Remark 2.2. From Theorem 2.3 we clearly see that

    (1) If n=1 and θ=0, then we get Theorem 2.4 in [72].

    (2) If V(ϱ)=n=1 and θ=0, then we get inequality (1.3) in [71].

    In the following result, the exponentially convex functions in Theorem 2.3 can be extended to exponentially quasi-convex functions.

    Theorem 2.4. Using the hypothesis of Theorem 2.2. If |Y| is θ-exponentially quasi-convex on I, then

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ| (2.12)
    (ρ2ρ1)n+1[max{|eθρ1Y(ρ1)|,|eθ(nρ1+ρ2n+1)Y(nρ1+ρ2n+1)|}
    +max{|eθρ2Y(ρ2)|,|eθ(ρ1+nρ2n+1)Y(ρ1+nρ2n+1)|}]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ

    for all nN.

    Proof. Using the exponentially quasi-convexity of |Y| for (2.6) in the proofs of Theorem 2.2, we get

    |Y(n+τn+1ρ1+1τn+1ρ2)|=max{|eθρ1Y(ρ1)|,|eθ(nρ1+ρ2n+1)Y(nρ1+ρ2n+1)|} (2.13)

    and

    |Y(1τn+1ρ1+n+τn+1ρ2)|=max{|eθρ2Y(ρ2)|,|eθ(ρ1+nρ2n+1)Y(ρ1+nρ2n+1)|}. (2.14)

    Combining (2.6), (2.13) and (2.14), we get the desired inequality (2.12).

    Next, we discuss some special cases of Theorem 2.4 as follows.

    Corollary 2.7. Let n=1. Then Theorem 2.4 reduces to

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(ρ1+ρ22)ρ2ρ1V(ϱ)dϱ|
    (ρ2ρ1)2[max{|eθρ1Y(ρ1)|,|eθ(ρ1+ρ22)Y(ρ1+ρ22)|}
    +max{|eθρ2Y(ρ2)|,|eθ(ρ1+ρ22)Y(ρ1+ρ22)|}]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    Corollary 2.8. Let θ=0. Then Theorem 2.4 leads to

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    (ρ2ρ1)n+1[max{|Y(ρ1)|,|Y(nρ1+ρ2n+1)|}
    +max{|Y(ρ2)|,|Y(ρ1+nρ2n+1)|}]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    Corollary 2.9. Let V(x)=1. Then Theorem 2.4 becomes

    |Y(nρ1+ρ2n+1)1ρ2ρ1ρ2ρ1Y(x)dx|
    (ρ2ρ1)2(n+1)[max{|Y(ρ1)|,|Y(nρ1+ρ2(n+1))|}
    +max{|Y(ρ2)|,|Y(ρ1+nρ2n+1)|}].

    Remark 2.3. If |Y| is increasing in Theorem 2.4, then

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ| (2.15)
    (ρ2ρ1)n+1[|eθρ2Y(ρ2)|+|eθ(ρ1+nρ2n+1)Y(ρ1+nρ2n+1)|]10L.(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ

    If |Y| is decreasing in Theorem 2.4, then

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ| (2.16)
    (ρ2ρ1)n+1[|eθρ1Y(ρ1)|+|eθ(nρ1+ρ2n+1)Y(nρ1+ρ2n+1)|]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    Remark 2.4. From Theorem 2.4 we clearly see that

    (1) Let n=1 and θ=0. Then Theorem 2.4 and Remark 2.3 lead to Theorem 2.8 and Remark 2.9 of [72], respectively.

    (2). Let n=V(ϱ)=1 and θ=0. Then we get Corollary 2.10 and Remark 2.11 of [72].

    Theorem 2.5. Suppose that all the hypothesis of Theorem 2.2 are satisfied, θR and λ1. If |Y|λ is θ-exponentially quasi-convex on I, then we have

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ| (2.17)
    (ρ2ρ1)n+1[(max{|eθρ1Y(ρ1)|λ,|eθ(nρ1+ρ2n+1)Y(nρ1+ρ2n+1)|λ})1λ
    +(max{|eθρ2Y(ρ2)|λ,|eθ(ρ1+nρ2n+1)Y(ρ1+nρ2n+1)|λ})1λ]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ

    for all nN.

    Proof. It follows from the exponentially quasi-convexity of |Y|λ and (2.6) that

    |Y(n+τn+1ρ1+1τn+1ρ2)|λmax{|eθρ1Y(ρ1)|λ,|eθ(nρ1+ρ2n+1)Y(nρ1+ρ2n+1)|λ} (2.18)

    and

    |Y(1τn+1ρ1+n+τn+1ρ2)|λmax{|eθρ2Y(ρ2)|λ,|eθ(ρ1+nρ2n+1)Y(ρ1+nρ2n+1)|λ}. (2.19)

    A combination of (2.6), (2.18) and (2.19) lead to the required inequality (2.17).

    Corollary 2.10. Let n=1. Then Theorem 2.5 reduces to

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(ρ1+ρ22)ρ2ρ1V(ϱ)dϱ|
    (ρ2ρ1)2[(max{|eθρ1Y(ρ1)|λ,|eθ(ρ1+ρ22)Y(ρ1+ρ22)|λ})1λ
    +(max{|eθρ2Y(ρ2)|λ,|eθ(ρ1+1ρ22)Y(ρ1+ρ22)|λ})1λ]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    Corollary 2.11. If θ=0, then Theorem 2.5 leads to the conclusion that

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    (ρ2ρ1)n+1[max{|Y(ρ1)|,|Y(nρ1+ρ2n+1)|}
    +max{|Y(ρ2)|,|Y(ρ1+nρ2n+1)|}]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ.

    In this section, we support our main results by presenting two examples.

    Example 3.1. Let ρ1=0, ρ2=π, θ=2, n=1, Y(ϱ)=sinϱ and V(ϱ)=cosϱ. Then all the assumptions in Theorem 2.2 are satisfied. Note that

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    =|π0sinϱcosϱdϱsinπ2π0cosϱdϱ|=1 (3.1)

    and

    ρ2ρ1n+1[|eθρ1Y(ρ1)|+|eθρ2Y(ρ2)|]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ
    =π2[|e0cos0)|+|e2πcosπ|]10L(0,π,τ)0cosϱdϱdτ
    =536.50π210(1τ)π20cosϱdϱdτ536.5. (3.2)

    From (3.1) and (3.2) we clearly Example 3.1 supports the conclusion of Theorem 2.2.

    Example 3.2. Let ρ1=0, ρ2=2, θ=0.5, n=2, Y(ϱ)=ϱ+2 and V(ϱ)=ϱ. Then all the assumptions in Theorem 2.2 are satisfied. Note that

    |ρ2ρ1Y(ϱ)V(ϱ)dϱY(nρ1+ρ2n+1)ρ2ρ1V(ϱ)dϱ|
    =|20ϱϱ+2dϱ8320ϱdϱ|0.3758 (3.3)

    and

    ρ2ρ1n+1[|eθρ1Y(ρ1)|+|eθρ2Y(ρ2)|]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ
    =23[|e0.5(0)122)|+|e0.5(2)14|]10L(0,2,τ)0ϱdϱdτ
    =0.6887102(1τ)30ϱdϱdτ1.0332. (3.4)

    From (3.3) and (3.4) we clearly see that Example 3.2 supports the conclusion of Theorem 2.2.

    Let Δ be a partition: ρ1=ϱ0<ϱ2<<ϱn1<ϱn=ρ2 of the interval [ρ1,ρ2] and consider the quadrature formula

    ρ2ρ1Y(ϱ)V(ϱ)dϱ=T(Y,V,p)+E(Y,V,p), (4.1)

    where

    T(Y,V,p)=κ1j=0Y(nϱj+ϱj+1n+1)ϱj+1ϱjV(ϱ)dϱ

    is weighted mean and E(Y,V,p) is the related approximation error.

    The aim of this subsection is to provide several new bounds for E(Y,V,p).

    Theorem 4.1. Let λ1, θR, and |Y|λ be θ-exponentially convex on I. Then the inequality

    |E(Y,V,p)|κ1j=0(ϱj+1ϱj)(|eθϱjY(ϱj)|λ+|eθϱj+1Y(ϱj+1)|λ2)1λ10L(ϱj,ϱj+1,τ)ϱjV(ϱ)dϱdτ

    holds for any pI if all the conditions of Theorem 2.2 are satisfied.

    Proof. Applying Theorem 2.3 to the interval [ϱj,ϱj+1] (j=0,1,...,κ1) of the partition Δ, we get

    |Y(nϱj+ϱj+1n+1)ϱj+1ϱjV(ϱ)dϱϱj+1ϱjY(ϱ)V(ϱ)dϱ|
    (ϱj+1ϱj)(|eθϱjY(ϱj)|λ+|eθϱj+1Y(ϱj+1)|λ2)1λ10L(ϱj,ϱj+1,τ)ϱjV(ϱ)dϱdτ.

    Summing the above inequality on j from 0 to κ1 and making use of the triangle inequality together with the exponential convexity of |Y|λ lead to

    |T(Y,V,p)ρ2ρ1Y(ϱ)V(ϱ)dϱ|
    κ1j=0(ϱj+1ϱj)(|eθϱjY(ϱj)|λ+|eθϱj+1Y(ϱj+1)|λ2)1λ10L(ϱj,ϱj+1,τ)ϱjV(ϱ)dϱdτ,

    this completes the proof of Theorem 4.1.

    Theorem 4.2. Let λ1, θR, and |Y|λ be θ-exponentially convex on I. Then the inequality

    |E(Y,V,p)|
    1n+1κ1j=0(ϱj+1ϱj)[[max{|eθϱjY(ϱj)|λ,|eθ(nϱj+ϱj+1n+1)Y(nϱj+ϱj+1n+1)|λ}]1λ
    +[max{|eθϱj+1Y(ϱj+1)|λ,|eθ(ϱj+nϱj+1n+1)τY(ϱj+nϱj+1n+1)|λ}]1λ]10L(ϱj,ϱj+1,τ)ϱjV(ϱ)dϱdτ

    holds for every partition Δ of I if all the hypothesis of Theorem 2.2 are satisfied.

    Proof. Making use of Theorem 2.5 on the interval [ϱj,ϱj+1] (j=0,1,,κ1) of the partition , we get

    |Y(nϱj+ϱj+1n+1)ϱj+1ϱjV(ϱ)dϱϱj+1ϱjY(ϱ)V(ϱ)dϱ|
    (ϱj+1ϱj)n+1[[max{|eθϱjY(ϱj)|λ,|eθ(nϱj+ϱj+1n+1)Y(nϱj+ϱj+1n+1)|λ}]1λ
    +[max{|eθϱj+1Y(ϱj+1)|λ,|eθ(ϱj+nϱj+1n+1)Y(ϱj+nϱj+1n+1)|λ}]1λ]10L(ϱj,ϱj+1,τ)ϱjV(ϱ)dϱdτ.

    Summing the above inequality on j from 0 to κ1 and making use the triangle inequality together with the exponential convexity of |Y|λ lead to the conclusion that

    |T(Y,V,p)ρ2ρ1Y(ϱ)V(ϱ)dϱ|
    1n+1κ1j=0(ϱj+1ϱj)[[max{|eθϱjY(ϱj)|λ,|eθ(nϱj+ϱj+1n+1)Y(nϱj+ϱj+1n+1)|λ}]1λ
    +[max{|eθϱj+1Y(ϱj+1)|λ,|eθ(ϱj+nϱj+1n+1)Y(ϱj+nϱj+1n+1)|λ}]1λ]10L(ϱj,ϱj+1,τ)ϱjV(ϱ)dϱdτ,

    this completes the proof of Theorem 4.2.

    Let 0<ρ1<ρ2, rR, V:[ρ1,ρ2][0,] be continuous on [ρ1,ρ2] and symmetric with respect to nρ1+ρ2n+1 and X be a continuous random variable having probability density function V. Then the rth-moment Er(X) of X is given by

    Er(X)=ρ2ρ1τrV(τ)dτ

    if it is finite.

    Theorem 4.3. The inequality

    |Er(X)(nρ1+ρ2n+1)r|r(ρ2ρ1)(n+1)2[|eθρ1ρr11|+|eθρ2ρr12|]

    holds for 0<ρ1<ρ2 and r2.

    Proof. Let Y(τ)=τr. Then |Y(τ)|=rτr1 is exponentially convex function. Note that

    ρ2ρ1Y(ϱ)V(ϱ)dϱ=Er(X),L(ρ1,ρ2,τ)ρ1V(ϱ)dϱnρ1+ρ2n+1ρ1V(ϱ)dϱ=1n+1(τ[0,1]),
    Y(nρ1+ρ2n+1)=(nρ1+ρ2n+1)r,|eθρ1Y(ρ1)|+|eθρ2Y(ρ2)|=r(eθρ1ρr11+eθρ2ρr12).

    Therefore, the desired result follows from inequality (2.2) immediately.

    Theorem 4.4. The inequality

    |Er(X)(nρ1+ρ2n+1)r|r(ρ2ρ1)(n+1)2[|eθρ2ρr12|+|eθ(nρ1+ρ2n+1)(nρ1+ρ2n+1)r1|]

    holds for 0<ρ1<ρ2 and r1.

    Proof. Let Y(τ)=τr. Then |Y(τ)|=rτr1 is increasing and exponentially quasi-convex, and the desired result can be obtained by use of inequality (2.15) and the similar arguments of Theorem 4.3.

    A real-valued function Ω:(0,)×(0,)(0,) is said to be a bivariate mean if min{ρ1,ρ2}Ω(ρ1,ρ2)max{ρ1,ρ2} for all ρ1,ρ2(0,). Recently, the properties and applications for the bivariate means and their related special functions have attracted the attention of many researchers [73,74,75,76,77,78,79,80,81,82,83,84,85,86]. In particular, many remarkable inequalities for the bivariate means can be found in the literature [87,88,89,90,91,92,93,94,95,96].

    In this subsection, we use the results obtained in Section 2 to give some applications to the special bivariate means.

    Let ρ1,ρ2>0 with ρ1ρ2. Then the arithmetic mean A(ρ1,ρ2), weighted arithmetic mean A(ρ1,ρ2;w1,w2) and n-th generalized logarithmic mean Ln(ρ1,ρ2) are defined by

    A(ρ1,ρ2)=ρ1+ρ12,A(ρ1,ρ2;w1,w2)=w1ρ1+w2ρ2w1+w2

    and

    Ln(ρ1,ρ2)=[ρn+12ρn+11(n+1)(ρ2ρ1)]1/n.

    Let ϱ>0, rN, Y(ϱ)=ϱr and V:[ρ1,ρ2]R+ be a differentiable mapping such that it is symmetric with respect to nρ1+ρ2n+1. Then Theorem 2.2 implies that

    |(nρ1+ρ2n+1)rρ2ρ1V(ϱ)dϱρ2ρ1ϱrV(ϱ)dϱ|r(ρ2ρ1)n+1[|eθρ1ρn11|+|eθρ2ρn12|]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ,

    which can be rewritten as

    |(A(ρ1,ρ2;n,1))rρ2ρ1V(ϱ)dϱρ2ρ1ϱrV(ϱ)dϱ|
    2r(ρ2ρ1)n+1[A(|eθρ1ρn11|,|eθρ2ρn12|)]10L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ. (4.2)

    Let V=1. Then inequality (4.2) leads to Corollary 4.1 immediately.

    Corollary 4.1. Let ρ2>ρ1>0, rN and r2. Then one has

    |(A(ρ1,ρ2;n,1))rLrr(ρ1,ρ2)|r(ρ2ρ1)2(n+1)2[A(|eθρ1ρn11|,|eθρ2ρn12|)].

    We conducted a preliminary attempt to develop a novel formulation presumably for new Hermite-Hadamard type for proposing two new classes of exponentially convex and exponentially quasi-convex functions and presented their analogues. An auxiliary result was chosen because of its success in leading to the well-known Hermite-Hadamard type inequalities. An intriguing feature of an auxiliary is that this simple formulation has significant importance while studying the error bounds of different numerical quadrature rules. Such a potential the connection needs further investigation. We conclude that the results derived in this paper are general in character and give some contributions to inequality theory and fractional calculus as an application for establishing the uniqueness of solutions in boundary value problems, fractional differential equations, and special relativity theory. This interesting aspect of time is worth further investigation. Finally, the innovative concept of exponentially convex functions has potential application in rth-moments and special bivariate mean to show the reported result. Our findings are the refinements and generalizations of the existing results that stimulate futuristic research.

    The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

    The research is supported by the Natural Science Foundation of China (Grant Nos. Grant Nos. 11701176, 61673169, 11301127, 11626101, 11601485).

    The authors declare that they have no competing interests.



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