Geometric characterization of the generalized Lommel–Wright function in the open unit disc

Abstract

Abstract 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 ) $\mathfrak{J}_{\lambda ,\mu}^{\nu ,m}(z):=\Gamma ^{m}(\lambda +1) \Gamma (\lambda +\mu +1)2^{2\lambda +\mu}z^{1-(\nu /2)-\lambda} \mathcal{J}_{\lambda ,\mu }^{\nu ,m}(\sqrt{z})$ , where the function J λ , μ ν , m $\mathcal{J}_{\lambda ,\mu}^{\nu ,m}$ 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 ) ) ′ $\mathcal{J}_{\lambda ,\mu}^{\nu ,m}(z):=(1-2\lambda -\nu )J_{ \lambda ,\mu}^{\nu ,m}(z)+z (J_{\lambda ,\mu }^{\nu ,m}(z) )^{\prime}$ 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 $$ J_{\nu ,\lambda}^{\mu ,m}(z)= \biggl(\frac{z}{2} \biggr)^{2\lambda + \nu} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{\Gamma ^{m} (k+\lambda +1 )\Gamma (k\mu +\nu +\lambda +1 )} \biggl(\frac{z}{ 2} \biggr)^{2k} $$ for λ ∈ C ∖ Z − $\lambda \in \mathbb{C}\setminus \mathbb{Z}^{-}$ , Z − : = { − 1 , − 2 , − 3 , … } $\mathbb{Z}^{-}:= \{ -1,-2,-3,\ldots \}$ , m ∈ N $m\in \mathbb{N}$ , ν ∈ C $\nu \in \mathbb{C}$ , and μ ∈ N 0 : = N ∪ { 0 } $\mu \in \mathbb{N}_{0}:=\mathbb{N}\cup \{0\}$ . 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 $0\leq \alpha <1$ . Ultimately, we discuss the starlikeness and convexity of order zero for J λ , μ ν , m $\mathfrak{J}_{\lambda ,\mu} ^{\nu ,m}$ , and it turns out that they are useful to extend the range of validity for the parameter λ to λ ≥ 0 $\lambda \geq 0$ 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.

Keywords

• Analytic
• Univalent
• Starlike
• Convex
• Convolution
• Pochhammer symbol