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 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 I⊆R be an interval, Y:I→R be a convex function. Then the double inequality
Y(ρ1+ρ22)≤1ρ2−ρ1ρ2∫ρ1Y(ϱ)dϱ≤Y(ρ1)+Y(ρ2)2 | (1.1) |
holds for all ρ1,ρ2∈I with ρ1≠ρ2. 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 θ∈R. Then a real-valued function Y:[0,∞)→R is said to be θ-exponentially convex if
Y(τρ1+(1−τ)ρ2)≤τeθρ1Y(ρ1)+(1−τ)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:R→R, 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 θ∈R. Then a mapping Y:[0,∞)⊆R→R is said to be θ-exponentially quasi-convex if
Y(τρ1+(1−τ)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 I⊆R be an interval, ρ1,ρ1∈I with ρ1<ρ2, and Y:I→R be a differentiable mapping on I∘ (where and in what follows I∘ denotes the interior of I) such that Y′∈L([ρ1,ρ2]) and |Y′| is convex on [ρ1,ρ2]. Then
|Y(ρ1+ρ22)−1ρ2−ρ1ρ2∫ρ1Y(ϱ)dϱ|≤(ρ2−ρ1)(|Y′(ρ1)|+|Y′(ρ2)|)8. | (1.3) |
Theorem 1.4. ([71]) Let λ∈R with λ≠0, I⊆R be an interval, ρ1,ρ1∈I with ρ1<ρ2, and Y:I→R be a differentiable mapping on I∘ such that Y′∈L([ρ1,ρ2]) and |Y′|λ is convex on [ρ1,ρ2]. Then
|Y(ρ1+ρ22)−1ρ2−ρ1ρ2∫ρ1Y(ϱ)dϱ|≤(ρ2−ρ1)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+1−τn+1ρ2 |
and
M(ρ1,ρ2,τ)=1−τn+1ρ1+n+τn+1ρ2 |
for τ∈[0,1] and all n∈N.
From now onwards, let ρ1,ρ2∈R 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 n∈N, Y:I→R be a differentiable mapping on I∘ such that Y′∈L1([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)]+ρ2−ρ12(n+1)1∫0{[Y(n+τn+1ρ1 |
1−τn+1ρ2)+Y(1−τn+1ρ1+n+τn+1ρ2)][U′(n+τn+1ρ1+1−τn+1ρ2)+U′(1−τn+1ρ1+n+τn+1ρ2)]}dτ |
=ρ2−ρ12(n+1){1∫0[U(n+τn+1ρ1+1−τn+1ρ2)−U(1−τn+1ρ1+n+τn+1ρ2)+U(ρ2)] |
×[−Y′(n+τn+1ρ1+1−τn+1ρ2)+Y′(1−τn+1ρ1+n+τn+1ρ2)]dτ}. | (2.1) |
Proof. It follows from integration by parts that
I1=−1∫0[U(n+τn+1ρ1+1−τn+1ρ2)−U(1−τn+1ρ1+n+τn+1ρ2)+U(ρ2)]Y′(n+τn+1ρ1+1−τn+1ρ2)dτ |
=n+1ρ2−ρ1{U(n+τn+1ρ1+1−τn+1ρ2)−U(1−τn+1ρ1+n+τn+1ρ2)+U(ρ2)}Y(n+τn+1ρ1+1−τn+1ρ2)|10 |
−ρ1−ρ2n+11∫0Y(n+τn+1ρ1+1−τn+1ρ2)[U′(n+τn+1ρ1+1−τn+1ρ2)+U′(1−τn+1ρ1+n+τn+1ρ2)]dτ |
=n+1ρ2−ρ1[U(ρ1)Y(ρ1)−[U(nρ1+ρ2n+1)−U(ρ1+nρ2n+1)+U(ρ2)]]Y(nρ1+ρ2n+1) |
+1∫0Y(n+τn+1ρ1+1−τn+1ρ2)[U′(n+τn+1ρ1+1−τn+1ρ2)+U′(1−τn+1ρ1+n+τn+1ρ2)]dτ. |
Similarly, we have
I2=1∫0[U(n+τn+1ρ1+1−τn+1ρ2)−U(1−τn+1ρ1+n+τn+1ρ2)+U(ρ2)]Y′(1−τn+1ρ1+n+τn+1ρ2)dτ |
=n+1ρ2−ρ1[U(ρ1)Y(ρ1)−[U(nρ1+ρ2n+1)−U(ρ1+nρ2n+1)+U(ρ2)]]Y(nρ1+ρ2n+1) |
+1∫0Y(1−τn+1ρ1+n+τn+1ρ2)[U′(n+τn+1ρ1+1−τn+1ρ2)+U′(1−τn+1ρ1+n+τn+1ρ2)]dτ. |
Adding I1 and I2, then multiplying by ρ2−ρ12(n+1) we get the desired identity (2.1).
Theorem 2.2. Let n∈N, θ∈R, Y:I→R 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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
≤ρ2−ρ1n+1[|eθρ1Y′(ρ1)|+|eθρ2Y′(ρ2)|]1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱdτ. | (2.2) |
Proof. Let τ∈[ρ1,ρ2] and Y(τ)=τ∫ρ1V(ϱ)dϱ. Then it follows from Lemma 2.1 that
ρ2−ρ12(n+1)1∫0[Y(n+τn+1ρ1+1−τn+1ρ2)+Y(1−τn+1ρ1+n+τn+1ρ2)][V(n+τn+1ρ1+1−τn+1ρ2) |
+V(1−τn+1ρ1+n+τn+1ρ2)]dτ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ |
=ρ2−ρ12(n+1)1∫0{L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱ+ρ2∫M(ρ1,ρ2,τ)V(ϱ)dϱ} |
×[−Y′(n+τn+1ρ1+1−τn+1ρ2)+Y′(1−τn+1ρ1+n+τn+1ρ2)]dτ. | (2.3) |
Since V(ϱ) is symmetric with respect to ϱ=nρ1+ρ2n+1, we have
ρ2−ρ12(n+1)1∫0[Y(n+τn+1ρ1+1−τn+1ρ2)+Y(1−τn+1ρ1+n+τn+1ρ2)][V(n+τn+1ρ1+1−τn+1ρ2) |
+V(1−τn+1ρ1+n+τn+1ρ2)]dτ |
=ρ2−ρ1(n+1)1∫0Y(n+τn+1ρ1+1−τn+1ρ2)V(n+τn+1ρ1+1−τn+1ρ2)dτ |
+ρ2−ρ1(n+1)1∫0Y(1−τn+1ρ1+n+τn+1ρ2)V(1−τn+1ρ1+n+τn+1ρ2)dτ |
=nρ1+ρ2n+1∫ρ1Y(ϱ)V(ϱ)dϱ+ρ2∫ρ1+nρ2n+1Y(ϱ)V(ϱ)dϱ=ρ2∫ρ1Y(ϱ)V(ϱ)dϱ | (2.4) |
and
L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱ=ρ2∫M(ρ1,ρ2,τ)V(ϱ)dϱ∀τ∈[0,1]. | (2.5) |
From (2.3)–(2.5) we clearly see that
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
≤ρ2−ρ1n+1{1∫0L(ρ1,ρ2,τ)∫ρ1|Y′(n+τn+1ρ1+1−τn+1ρ2)|dτ+1∫0L(ρ1,ρ2,τ)∫ρ1|Y′(1−τn+1ρ1+n+τn+1ρ2)|dτ}. | (2.6) |
Making use of the exponentially convexity of |Y′| we get
1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)|Y′(n+τn+1ρ1+1−τn+1ρ2)|dϱdτ+1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)|Y′(1−τn+1ρ1+n+τn+1ρ2)|dϱdτ |
≤1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)[n+τn+1|eθρ1Y′(ρ1)|+1−τn+1|eθρ2Y′(ρ2)|+1−τn+1|eθρ1Y′(ρ1)+n+τn+1|eθρ2Y′(ρ2)||]dϱdτ |
=[|eθρ1Y′(ρ1)|+|eθρ2Y′(ρ2)|]1∫0L(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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
≤ρ2−ρ1n+1[|Y′(ρ1)|+|Y′(ρ2)|]1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱdτ. |
Corollary 2.2. Let n=1. Then Theorem 2.2 reduces to
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(ρ1+ρ22)ρ2∫ρ1V(ϱ)dϱ| |
≤ρ2−ρ12[|eθρ1Y′(ρ1)|+|eθρ2Y′(ρ2)|]1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱdτ. |
Corollary 2.3. Let V(ϱ)=1. Then then Theorem 2.3 becomes
|Y(nρ1+ρ2n+1)−1ρ2−ρ1ρ2∫ρ1Y(ϱ)dϱ| |
≤ρ2−ρ12(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 λ≥1. If θ∈R and |Y′|λ is θ-exponentially convex on I, then
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
≤2(ρ2−ρ1)n+1[|eθρ1Y′(ρ1)|λ+|eθρ2Y′(ρ2)|λ2]1λ1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ | (2.8) |
for all n∈N.
Proof. Continuing inequality (2.6) in the proofs of Theorem 2.2 and using the well-known Hölder integral inequality, one has
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
≤ρ2−ρ1n+1{(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ)1−1λ(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)|Y′(n+τn+1ρ1+1−τn+1ρ2)|λdϱdτ)1λ |
+(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ)1−1λ(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)|Y′(1−τn+1ρ1+n+τn+1ρ2)|λdϱdτ)1λ} |
≤ρ2−ρ1n+1(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ)1−1λ{(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)|Y′(n+τn+1ρ1+1−τn+1ρ2)|λdϱdτ)1λ |
+(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)|Y′(1−τn+1ρ1+n+τn+1ρ2)|λdϱdτ)1λ}. | (2.9) |
It follows from the power-mean inequality
μa+νa<21−a(μ+ν)a |
for μ,ν>0 and a<1 that
(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)|Y′(n+τn+1ρ1+1−τn+1ρ2)|λdϱdτ)1λ | (2.10) |
+(1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)|Y′(1−τn+1ρ1+n+τn+1ρ2)|λdϱdτ)1λ |
≤21−1λ{1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)(|Y′(n+τn+1ρ1+1−τn+1ρ2)|λ+|Y′(1−τn+1ρ1+n+τn+1ρ2)|λ)dϱdτ}1λ. |
Since |Y′|λ is an θ-exponentially convex on I, we have
|Y′(n+τn+1ρ1+1−τn+1ρ2)|λ+|Y′(1−τn+1ρ1+n+τn+1ρ2)| |
≤n+τn+1|eθρ1Y′(ρ1)|q+1−τn+1|eθρ2Y′(ρ2)|q+1−τn+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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(ρ1+ρ22)ρ2∫ρ1V(ϱ)dϱ| |
≤(ρ2−ρ1)[|eθρ1Y′(ρ1)|λ+|eθρ2Y′(ρ2)|λ2]1λ1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ. |
Corollary 2.5. Let θ=0. Then Theorem 2.3 leads to
|ρ2∫ρ1Y(x)V(x)dx−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
≤2(ρ2−ρ1)n+1[|Y′(ρ1)|λ+|Y′(ρ2)|λ2]1λ1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ. |
Corollary 2.6. Let V(ϱ)=1. Then Theorem 2.3 becomes
|Y(nρ1+ρ2n+1)−1ρ2−ρ1ρ2∫ρ1Y(ϱ)dϱ|≤(ρ2−ρ1)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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| | (2.12) |
≤(ρ2−ρ1)n+1[max{|eθρ1Y′(ρ1)|,|eθ(nρ1+ρ2n+1)Y′(nρ1+ρ2n+1)|} |
+max{|eθρ2Y′(ρ2)|,|eθ(ρ1+nρ2n+1)Y′(ρ1+nρ2n+1)|}]1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ |
for all n∈N.
Proof. Using the exponentially quasi-convexity of |Y′| for (2.6) in the proofs of Theorem 2.2, we get
|Y′(n+τn+1ρ1+1−τn+1ρ2)|=max{|eθρ1Y′(ρ1)|,|eθ(nρ1+ρ2n+1)Y′(nρ1+ρ2n+1)|} | (2.13) |
and
|Y′(1−τn+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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(ρ1+ρ22)ρ2∫ρ1V(ϱ)dϱ| |
≤(ρ2−ρ1)2[max{|eθρ1Y′(ρ1)|,|eθ(ρ1+ρ22)Y′(ρ1+ρ22)|} |
+max{|eθρ2Y′(ρ2)|,|eθ(ρ1+ρ22)Y′(ρ1+ρ22)|}]1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ. |
Corollary 2.8. Let θ=0. Then Theorem 2.4 leads to
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
≤(ρ2−ρ1)n+1[max{|Y′(ρ1)|,|Y′(nρ1+ρ2n+1)|} |
+max{|Y′(ρ2)|,|Y′(ρ1+nρ2n+1)|}]1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ. |
Corollary 2.9. Let V(x)=1. Then Theorem 2.4 becomes
|Y(nρ1+ρ2n+1)−1ρ2−ρ1ρ2∫ρ1Y(x)dx| |
≤(ρ2−ρ1)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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| | (2.15) |
≤(ρ2−ρ1)n+1[|eθρ2Y′(ρ2)|+|eθ(ρ1+nρ2n+1)Y′(ρ1+nρ2n+1)|]1∫0∫L.(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ |
If |Y′| is decreasing in Theorem 2.4, then
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| | (2.16) |
≤(ρ2−ρ1)n+1[|eθρ1Y′(ρ1)|+|eθ(nρ1+ρ2n+1)Y′(nρ1+ρ2n+1)|]1∫0∫L(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, θ∈R and λ≥1. If |Y′|λ is θ-exponentially quasi-convex on I, then we have
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| | (2.17) |
≤(ρ2−ρ1)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]1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ |
for all n∈N.
Proof. It follows from the exponentially quasi-convexity of |Y′|λ and (2.6) that
|Y′(n+τn+1ρ1+1−τn+1ρ2)|λ≤max{|eθρ1Y′(ρ1)|λ,|eθ(nρ1+ρ2n+1)Y′(nρ1+ρ2n+1)|λ} | (2.18) |
and
|Y′(1−τn+1ρ1+n+τn+1ρ2)|λ≤max{|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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(ρ1+ρ22)ρ2∫ρ1V(ϱ)dϱ| |
≤(ρ2−ρ1)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]1∫0∫L(ρ1,ρ2,τ)ρ1V(ϱ)dϱdτ. |
Corollary 2.11. If θ=0, then Theorem 2.5 leads to the conclusion that
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
≤(ρ2−ρ1)n+1[max{|Y′(ρ1)|,|Y′(nρ1+ρ2n+1)|} |
+max{|Y′(ρ2)|,|Y′(ρ1+nρ2n+1)|}]1∫0∫L(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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
=|π∫0sinϱcosϱdϱ−sinπ2π∫0cosϱdϱ|=1 | (3.1) |
and
ρ2−ρ1n+1[|eθρ1Y′(ρ1)|+|eθρ2Y′(ρ2)|]1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱdτ |
=π2[|e0cos0)|+|e2πcosπ|]1∫0L(0,π,τ)∫0cosϱdϱdτ |
=536.50π21∫0(1−τ)π2∫0cosϱdϱdτ≈536.5. | (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
|ρ2∫ρ1Y(ϱ)V(ϱ)dϱ−Y(nρ1+ρ2n+1)ρ2∫ρ1V(ϱ)dϱ| |
=|2∫0ϱ√ϱ+2dϱ−√832∫0ϱdϱ|≈0.3758 | (3.3) |
and
ρ2−ρ1n+1[|eθρ1Y′(ρ1)|+|eθρ2Y′(ρ2)|]1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱdτ |
=23[|e0.5(0)12√2)|+|e0.5(2)14|]1∫0L(0,2,τ)∫0ϱdϱdτ |
=0.68871∫02(1−τ)3∫0ϱdϱdτ≈1.0332. | (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<⋯<ϱn−1<ϱn=ρ2 of the interval [ρ1,ρ2] and consider the quadrature formula
ρ2∫ρ1Y(ϱ)V(ϱ)dϱ=T(Y,V,p)+E(Y,V,p), | (4.1) |
where
T(Y,V,p)=κ−1∑j=0Y(nϱj+ϱj+1n+1)ϱj+1∫ϱjV(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 λ≥1, θ∈R, and |Y′|λ be θ-exponentially convex on I. Then the inequality
|E(Y,V,p)|≤κ−1∑j=0(ϱj+1−ϱj)(|eθϱjY′(ϱj)|λ+|eθϱj+1Y′(ϱj+1)|λ2)1λ1∫0L(ϱj,ϱj+1,τ)∫ϱjV(ϱ)dϱdτ |
holds for any p∈I if all the conditions of Theorem 2.2 are satisfied.
Proof. Applying Theorem 2.3 to the interval [ϱj,ϱj+1] (j=0,1,...,κ−1) of the partition Δ, we get
|Y(nϱj+ϱj+1n+1)ϱj+1∫ϱjV(ϱ)dϱ−ϱj+1∫ϱjY(ϱ)V(ϱ)dϱ| |
≤(ϱj+1−ϱj)(|eθϱjY′(ϱj)|λ+|eθϱj+1Y′(ϱj+1)|λ2)1λ1∫0L(ϱj,ϱj+1,τ)∫ϱjV(ϱ)dϱdτ. |
Summing the above inequality on j from 0 to κ−1 and making use of the triangle inequality together with the exponential convexity of |Y′|λ lead to
|T(Y,V,p)−ρ2∫ρ1Y(ϱ)V(ϱ)dϱ| |
≤κ−1∑j=0(ϱj+1−ϱj)(|eθϱjY′(ϱj)|λ+|eθϱj+1Y′(ϱj+1)|λ2)1λ1∫0L(ϱj,ϱj+1,τ)∫ϱjV(ϱ)dϱdτ, |
this completes the proof of Theorem 4.1.
Theorem 4.2. Let λ≥1, θ∈R, and |Y′|λ be θ-exponentially convex on I. Then the inequality
|E(Y,V,p)| |
≤1n+1κ−1∑j=0(ϱj+1−ϱj)[[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]1∫0L(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,⋯,κ−1) of the partition △, we get
|Y(nϱj+ϱj+1n+1)ϱj+1∫ϱjV(ϱ)dϱ−ϱj+1∫ϱjY(ϱ)V(ϱ)dϱ| |
≤(ϱj+1−ϱj)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]1∫0L(ϱj,ϱj+1,τ)∫ϱjV(ϱ)dϱdτ. |
Summing the above inequality on j from 0 to κ−1 and making use the triangle inequality together with the exponential convexity of |Y′|λ lead to the conclusion that
|T(Y,V,p)−ρ2∫ρ1Y(ϱ)V(ϱ)dϱ| |
≤1n+1κ−1∑j=0(ϱj+1−ϱj)[[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]1∫0L(ϱj,ϱj+1,τ)∫ϱjV(ϱ)dϱdτ, |
this completes the proof of Theorem 4.2.
Let 0<ρ1<ρ2, r∈R, 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)=ρ2∫ρ1τrV(τ)dτ |
if it is finite.
Theorem 4.3. The inequality
|Er(X)−(nρ1+ρ2n+1)r|≤r(ρ2−ρ1)(n+1)2[|eθρ1ρr−11|+|eθρ2ρr−12|] |
holds for 0<ρ1<ρ2 and r≥2.
Proof. Let Y(τ)=τr. Then |Y′(τ)|=rτr−1 is exponentially convex function. Note that
ρ2∫ρ1Y(ϱ)V(ϱ)dϱ=Er(X),L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱ≤nρ1+ρ2n+1∫ρ1V(ϱ)dϱ=1n+1(τ∈[0,1]), |
Y(nρ1+ρ2n+1)=(nρ1+ρ2n+1)r,|eθρ1Y′(ρ1)|+|eθρ2Y′(ρ2)|=r(eθρ1ρr−11+eθρ2ρr−12). |
Therefore, the desired result follows from inequality (2.2) immediately.
Theorem 4.4. The inequality
|Er(X)−(nρ1+ρ2n+1)r|≤r(ρ2−ρ1)(n+1)2[|eθρ2ρr−12|+|eθ(nρ1+ρ2n+1)(nρ1+ρ2n+1)r−1|] |
holds for 0<ρ1<ρ2 and r≥1.
Proof. Let Y(τ)=τr. Then |Y(τ)|=rτr−1 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 ρ1≠ρ2. 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+12−ρn+11(n+1)(ρ2−ρ1)]1/n. |
Let ϱ>0, r∈N, 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)rρ2∫ρ1V(ϱ)dϱ−ρ2∫ρ1ϱrV(ϱ)dϱ|≤r(ρ2−ρ1)n+1[|eθρ1ρn−11|+|eθρ2ρn−12|]1∫0L(ρ1,ρ2,τ)∫ρ1V(ϱ)dϱdτ, |
which can be rewritten as
|(A(ρ1,ρ2;n,1))rρ2∫ρ1V(ϱ)dϱ−ρ2∫ρ1ϱrV(ϱ)dϱ| |
≤2r(ρ2−ρ1)n+1[A(|eθρ1ρn−11|,|eθρ2ρn−12|)]1∫0L(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, r∈N and r≥2. Then one has
|(A(ρ1,ρ2;n,1))r−Lrr(ρ1,ρ2)|≤r(ρ2−ρ1)2(n+1)2[A(|eθρ1ρn−11|,|eθρ2ρn−12|)]. |
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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