On some geometric properties for the combination of generalized Lommel–Wright function

Journal of Inequalities and Applications

Hanaa M. Zayed¹  Teodor Bulboacă² 

[1] Department of Mathematics and Computer Science, Faculty of Science, Menoufia University;Faculty of Mathematics and Computer Science, Babeş-Bolyai University;

关键词: Analytic;    Univalent;    Starlike;    Convex;    Close-to-convex;    Generalized Lommel-Wright functions;   

DOI  :  10.1186/s13660-021-02690-z

来源: DOAJ

【 摘 要 】

Abstract 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 $J_{\nu ,\lambda }^{\mu ,m}$ 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 ) $\mathfrak{J}_{\nu ,\lambda }^{\mu ,m}(z):=\Gamma ^{m}(\lambda +1) \Gamma (\lambda +\nu +1)2^{2\lambda +\nu } z^{1-(\nu /2)-\lambda } \mathcal{I}_{\nu ,\lambda }^{\mu ,m}(\sqrt{z})$ , where I ν , λ μ , m ( z ) : = ( 1 − 2 λ − ν ) J ν , λ μ , m ( z ) + z ( J ν , λ μ , m ( z ) ) ′ $\mathcal{I}_{\nu ,\lambda }^{\mu ,m}(z):=(1-2\lambda -\nu )J_{\nu , \lambda }^{\mu ,m}(z)+z (J_{\nu ,\lambda }^{\mu ,m}(z) )^{ \prime }$ 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 , $$ J_{\nu ,\lambda }^{\mu ,m}(z)= \biggl(\frac{z}{2} \biggr)^{2\lambda + \nu } \sum_{n=0}^{\infty } \frac{(-1)^{n}}{\Gamma ^{m} (n+\lambda +1 )\Gamma (n\mu +\nu +\lambda +1 )} \biggl(\frac{z}{2} \biggr)^{2n}, $$ with m ∈ N $m\in \mathbb{N}$ , μ > 0 $\mu >0$ and λ , ν ∈ C $\lambda ,\nu \in \mathbb{C}$ , including starlikeness and convexity of order α ( 0 ≤ α < 1 $0\leq \alpha <1$ ) 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 $\lambda >-1$ then Re ( J ν , λ μ , m ( z ) / z ) > α $\operatorname{Re} (\mathfrak{J}_{\nu ,\lambda }^{\mu ,m}(z)/z )>\alpha $ , z ∈ U $z\in \mathbb{U}$ , and if λ ≥ ( 10 − 6 ) / 4 $\lambda \ge (\sqrt{10}-6 )/4$ then the function ( J ν , λ μ , m ( z 2 ) / z ) ∗ sin z $(\mathfrak{J}_{\nu ,\lambda }^{\mu ,m}(z^{2})/z )\ast \sin z$ is close-to-convex with respect to 1 / 2 log ( ( 1 + z ) / ( 1 − z ) ) $1/2\log ((1+z)/(1-z) )$ where ∗ stands for the Hadamard product (or convolution) of two power series.

【 授权许可】

Unknown