Generalized Lommel–Wright function and its geometric properties

Abstract

Abstract The normalization of the combination of generalized Lommel–Wright function J κ 1 , κ 2 κ 3 , m ( z ) $\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)$ ( m ∈ N , κ 3 > 0 $\kappa _{3}>0$ 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 ) $\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z):=\Gamma ^{m}( \kappa _{1}+1)\Gamma (\kappa _{1}+\kappa _{2}+1)2^{2\kappa _{1}+ \kappa _{2}}z^{1-(\kappa _{2}/2)-\kappa _{1}}\mathcal{J}_{\kappa _{1},\kappa _{2}}^{ \kappa _{3},m}(\sqrt{z})$ , 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 ) ) ′ $\mathcal{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z):=(1-2\kappa _{1}-\kappa _{2})J_{\kappa _{1},\kappa _{2}}^{ \kappa _{3},m}(z)+z ( J_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z) ) ^{\prime}$ 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 , $$ J_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)= \biggl( \frac{z}{2} \biggr) ^{2\kappa _{1}+\kappa _{2}}\sum_{n=0}^{\infty} \frac{(-1)^{n}}{\Gamma ^{m} ( n+\kappa _{1}+1 ) \Gamma ( n\kappa _{3}+\kappa _{1}+\kappa _{2}+1 ) } \biggl( \frac{z}{2} \biggr) ^{2n}, $$ 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 ) $\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)$ to be starlike and convex of order α, 0 ≤ α < 1 $0\leq \alpha <1$ , 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 ) $\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)$ defined by I κ 1 , κ 2 κ 3 , m ( z ) = J κ 1 , κ 2 κ 3 , m ( z ) ∗ z / ( 1 + z ) $\mathcal{I}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)=\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)\ast 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 ) $\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)\ $ is underway and will be presented in a forthcoming paper.

Keywords

• Analytic
• Univalent
• Starlike
• Convex
• Generalized Lommel–Wright functions