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

Abstract The scope of our investigation is to study the geometric properties normalized form combination generalized Lommel–Wright function $J_{\nu ,\lambda }^{\mu ,m}$ J ? , ? ? m defined by $\mathfrak{J}_{\nu ,m}(z):=\Gamma ^{m}(\lambda +1) \Gamma (\lambda +\nu +1)2^{2\lambda } z^{1-(\nu /2)-\lambda \mathcal{I}_{\nu ,m}(\sqrt{z})$ ( z ) : = ? + 1 2 ? / I , where $\mathcal{I}_{\nu ,m}(z):=(1-2\lambda -\nu )J_{\nu \lambda ,m}(z)+z (J_{\nu ,m}(z) )^{ \prime }$ ? and $$ J_{\nu ,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 +\lambda )} \biggr)^{2n}, ? n 0 ? with $m\in \mathbb{N}$ ? N $\mu >0$ > $\lambda ,\nu \in \mathbb{C}$ C including starlikeness convexity order ? ( $0\leq \alpha <1$ ? ? < ) in open unit disc using two-sided inequality for Fox–Wright functions that has been proved Pogány Srivastava (Comput. Math. Appl. 57(1):127–140, 2009). Further, orders are also evaluated some classical tools. We then compare given both techniques illustrate efficacy approach. In addition, we values if >-1$ $\operatorname{Re} (\mathfrak{J}_{\nu ,m}(z)/z )>\alpha $ Re $z\in \mathbb{U}$ U \ge (\sqrt{10}-6 )/4$ ? 10 6 4 $(\mathfrak{J}_{\nu ,m}(z^{2})/z )\ast \sin z$ ? sin close-to-convex respect $1/2\log ((1+z)/(1-z) )$ log ? stands Hadamard product (or convolution) two power series.