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

Abstract

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 $\lambda >-1$ 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.

Keywords

• Analytic
• Univalent
• Starlike
• Convex
• Close-to-convex
• Generalized Lommel-Wright functions