| Popis: |
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 |
