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The normalization of the combination of generalized Lommel–Wright function J κ 1 , κ 2 κ 3 , m (z) (m ∈ N , κ 3 > 0 and κ 1 , κ 2 ∈ C ) defined by J κ 1 , κ 2 κ 3 , m (z) : = Γ m (κ 1 + 1) Γ (κ 1 + κ 2 + 1) 2 2 κ 1 + κ 2 z 1 − (κ 2 / 2) − κ 1 J κ 1 , κ 2 κ 3 , m (z) , where J κ 1 , κ 2 κ 3 , m (z) : = (1 − 2 κ 1 − κ 2) J κ 1 , κ 2 κ 3 , m (z) + z (J κ 1 , κ 2 κ 3 , m (z)) ′ and J κ 1 , κ 2 κ 3 , m (z) = (z 2) 2 κ 1 + κ 2 ∑ n = 0 ∞ (− 1) n Γ m (n + κ 1 + 1) Γ (n κ 3 + κ 1 + κ 2 + 1) (z 2) 2 n , was previously introduced and some of its geometric properties have been considered. In this paper, we report conditions for J κ 1 , κ 2 κ 3 , m (z) to be starlike and convex of order α, 0 ≤ α < 1 , inside the open unit disk using some technical manipulations of the gamma and digamma functions, as well as inequality for the digamma function that has been proved (Guo and Qi in Proc. Am. Math. Soc. 141(3):1007–1015, 2013). In addition, a method presented by Lorch (J. Approx. Theory 40(2):115–120 1984) and further developed by Laforgia (Math. Compet. 42(166):597–600 1984) is applied to establish firstly sharp inequalities for the shifted factorial that will be used to obtain the order of starlikeness and convexity. We compare then the obtained orders of starlikeness and convexity with some important consequences in the literature as well as the results proposed by all techniques to demonstrate the accuracy of our approach. Ultimately, a lemma by (Fejér in Acta Litt. Sci. 8:89–115 1936) is used to prove that the modified form of the function J κ 1 , κ 2 κ 3 , m (z) defined by I κ 1 , κ 2 κ 3 , m (z) = J κ 1 , κ 2 κ 3 , m (z) ∗ z / (1 + z) is in the class of starlike and convex functions, respectively. Further work regarding the function J κ 1 , κ 2 κ 3 , m (z) is underway and will be presented in a forthcoming paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider a distributed optimization problem of minimizing sum of convex functions over the intersection of fixed-point constraints. We propose a distributed method for solving the problem. We prove the convergence of the generated sequence to the solution of the problem under certain assumption. We further discuss the convergence rate with an appropriate positive stepsize. A numerical experiment is given to show the effectiveness of the obtained theoretical result. [ABSTRACT FROM AUTHOR]

The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function J λ , μ ν , m (z) : = Γ m (λ + 1) Γ (λ + μ + 1) 2 2 λ + μ z 1 − (ν / 2) − λ J λ , μ ν , m (z) , where the function J λ , μ ν , m satisfies the differential equation J λ , μ ν , m (z) : = (1 − 2 λ − ν) J λ , μ ν , m (z) + z (J λ , μ ν , m (z)) ′ with J ν , λ μ , m (z) = (z 2) 2 λ + ν ∑ k = 0 ∞ (− 1) k Γ m (k + λ + 1) Γ (k μ + ν + λ + 1) (z 2) 2 k for λ ∈ C ∖ Z − , Z − : = { − 1 , − 2 , − 3 , ... } , m ∈ N , ν ∈ C , and μ ∈ N 0 : = N ∪ { 0 } . In particular, we employ a new procedure using mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by Li and Chen (J. Inequal. Pure Appl. Math. 8(1):28, 2007), to evaluate the starlikeness and convexity of order α, 0 ≤ α < 1 . Ultimately, we discuss the starlikeness and convexity of order zero for J λ , μ ν , m , and it turns out that they are useful to extend the range of validity for the parameter λ to λ ≥ 0 where the main concept of the proofs comes from some technical manipulations given by Mocanu (Libertas Math. 13:27–40, 1993). Our results improve, complement, and generalize some well-known (nonsharp) estimates. [ABSTRACT FROM AUTHOR]

We consider the extremal problem of maximizing functions u in the class of real-valued biconvex functions satisfying a boundary condition ψ on a product of the unit ball with itself, with the $\ell^{p}$ -norm. In 1986, Burkholder explicitly found the maximal function for $p=2$ . In this paper, we find some characterizations of such extremal functions. We establish that sufficiently smooth solutions to the convex extremal problems with given boundary values are affine on line segments and the domain D is foliated by such segments. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to derive a new post quantum integral identity using twice (p , q) -differentiable functions. Using this identity as an auxiliary result, we obtain some new post quantum estimates of upper bounds involving twice (p , q) -differentiable preinvex functions. [ABSTRACT FROM AUTHOR]

The scope of our investigation is to study the geometric properties of the normalized form of the combination of generalized Lommel–Wright function J ν , λ μ , m defined by J ν , λ μ , m (z) : = Γ m (λ + 1) Γ (λ + ν + 1) 2 2 λ + ν z 1 − (ν / 2) − λ I ν , λ μ , m (z) , where I ν , λ μ , m (z) : = (1 − 2 λ − ν) J ν , λ μ , m (z) + z (J ν , λ μ , m (z)) ′ and J ν , λ μ , m (z) = (z 2) 2 λ + ν ∑ n = 0 ∞ (− 1) n Γ m (n + λ + 1) Γ (n μ + ν + λ + 1) (z 2) 2 n , with m ∈ N , μ > 0 and λ , ν ∈ C , including starlikeness and convexity of order α (0 ≤ α < 1 ) in the open unit disc using the two-sided inequality for the Fox–Wright functions that has been proved by Pogány and Srivastava in (Comput. Math. Appl. 57(1):127–140, 2009). Further, the orders of starlikeness and convexity are also evaluated using some classical tools. We then compare the orders of starlikeness and convexity given by both techniques to illustrate the efficacy of the approach. In addition, we proved that for some values of α, if λ > − 1 then Re (J ν , λ μ , m (z) / z) > α , z ∈ U , and if λ ≥ (10 − 6) / 4 then the function (J ν , λ μ , m (z 2) / z) ∗ sin z is close-to-convex with respect to 1 / 2 log ((1 + z) / (1 − z)) where ∗ stands for the Hadamard product (or convolution) of two power series. [ABSTRACT FROM AUTHOR]

The main objective of this article is to establish a new post quantum version of Montgomery identity. Some estimates of associated post quantum bounds are also obtained. In order to obtain the main results of the article, we use the preinvexity property of the functions. Some special cases are also discussed in detail. Finally, we present some applications of the obtained results, which shows the significance of the discussed results. [ABSTRACT FROM AUTHOR]

We prove several relations of the type ∣arg{zf''(z)/f'(z)}∣ ≤ ∣arg{f'(z)}∣ for functions satisfying some geometric conditions. [ABSTRACT FROM AUTHOR]