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

The scope of our investigation is to study the geometric properties of the normalized form of the combination of generalized Lommel-Wright function J(nu,lambda)(mu,m) defined by J(nu,lambda)(mu,m) (z) := Gamma(m)(lambda+ 1)Gamma(lambda +nu + 1)2(2 lambda)+(nu)z(1-(nu/2))-lambda I-nu,lambda(mu,m) (root z), where I-nu,lambda(mu,m) (z) := (1 - 2 lambda -nu)J(nu lambda)(mu,m.)(z) + z(J(nu),(mu,m.)(lambda) (z))' and J(nu,lambda)(mu,m) (z) = (z/2)2 lambda+nu Sigma(infinity)(n=0)(-1)n/Gamma m(n+lambda+1)Gamma(n mu + nu + lambda +1)(z/2)(2n), with m is an element of N, mu > 0 and lambda, nu is an element of C, including starlikeness and convexity of order alpha (0 <= alpha < 1) in the open unit disc using the two-sided inequality for the Fox-Wright functions that has been proved by Pogany and Srivastava in (Comput. Math. Appl. 57(1):127-140, 2009). Further, the orders of starlikeness and convexity are also evaluated using some classical tools. We then compare the orders of starlikeness and convexity given by both techniques to illustrate the efficacy of the approach. In addition, we proved that for some values of a, if lambda > -1 then Re(J(nu,lambda)(mu,m) (z)/z) > a, z is an element of U, and if lambda = (root 10 - 6)/4 then the function (J(nu,lambda)(mu,m) (z(2))/z) * sin z is close-to-convex with respect to 1/2 log((1 + z)/(1 - z)) where * stands for the Hadamard product (or convolution) of two power series.