Generalized Lommel–Wright function and its geometric properties

Abstract

AbstractThe normalization of the combination of generalized Lommel–Wright function$\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)$Jκ1,κ2κ3,m(z)(m∈N,$\kappa _{3}>0$κ3>0andκ1,κ2∈C) defined by$\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z):=\Gamma ^{m}( \kappa _{1}+1)\Gamma (\kappa _{1}+\kappa _{2}+1)2^{2\kappa _{1}+ \kappa _{2}}z^{1-(\kappa _{2}/2)-\kappa _{1}}\mathcal{J}_{\kappa _{1},\kappa _{2}}^{ \kappa _{3},m}(\sqrt{z})$Jκ1,κ2κ3,m(z):=Γm(κ1+1)Γ(κ1+κ2+1)22κ1+κ2z1−(κ2/2)−κ1Jκ1,κ2κ3,m(z), where$\mathcal{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z):=(1-2\kappa _{1}-\kappa _{2})J_{\kappa _{1},\kappa _{2}}^{ \kappa _{3},m}(z)+z ( J_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z) ) ^{\prime}$Jκ1,κ2κ3,m(z):=(1−2κ1−κ2)Jκ1,κ2κ3,m(z)+z(Jκ1,κ2κ3,m(z))′and$$ J_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)= \biggl( \frac{z}{2} \biggr) ^{2\kappa _{1}+\kappa _{2}}\sum_{n=0}^{\infty} \frac{(-1)^{n}}{\Gamma ^{m} ( n+\kappa _{1}+1 ) \Gamma ( n\kappa _{3}+\kappa _{1}+\kappa _{2}+1 ) } \biggl( \frac{z}{2} \biggr) ^{2n}, $$Jκ1,κ2κ3,m(z)=(z2)2κ1+κ2∑n=0∞(−1)nΓm(n+κ1+1)Γ(nκ3+κ1+κ2+1)(z2)2n,was previously introduced and some of its geometric properties have been considered. In this paper, we report conditions for$\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)$Jκ1,κ2κ3,m(z)to be starlike and convex of orderα,$0\leq \alpha <1$0≤α<1, inside the open unit disk using some technical manipulations of the gamma and digamma functions, as well as inequality for the digamma function that has been proved (Guo and Qi in Proc. Am. Math. Soc. 141(3):1007–1015, 2013). In addition, a method presented by Lorch (J. Approx. Theory 40(2):115–120 1984) and further developed by Laforgia (Math. Compet. 42(166):597–600 1984) is applied to establish firstly sharp inequalities for the shifted factorial that will be used to obtain the order of starlikeness and convexity. We compare then the obtained orders of starlikeness and convexity with some important consequences in the literature as well as the results proposed by all techniques to demonstrate the accuracy of our approach. Ultimately, a lemma by (Fejér in Acta Litt. Sci. 8:89–115 1936) is used to prove that the modified form of the function$\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)$Jκ1,κ2κ3,m(z)defined by$\mathcal{I}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)=\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)\ast z/(1+z) \ $Iκ1,κ2κ3,m(z)=Jκ1,κ2κ3,m(z)∗z/(1+z)is in the class of starlike and convex functions, respectively. Further work regarding the function$\mathfrak{J}_{\kappa _{1},\kappa _{2}}^{\kappa _{3},m}(z)\ $Jκ1,κ2κ3,m(z)is underway and will be presented in a forthcoming paper.