Geometrical and Computational Properties of the Generalized Struve Functions

The normalization of the generalized Struve functions Hρ,r(z)(ρ,r∈C) defined by (Formula presented.) was introduced previously and some of its geometric properties have been presented in Orhan and Yağmur (An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (NS) 63(2):229–244, 2017). In this paper, we determine conditions for Hρ,r(z) to be starlike and convex of different orders within the open unit disk using inequalities for the digamma function and its derivative that have been proved in Gu and Qi (J. Approx. Theory 163(9):1208–1216, 2011) as well as other preliminary lemmas in Fejér (Acta Litterarum ac Scientiarum 8:89–115, 1936) and Ozaki (Sci. Rep. Tokyo Bunrika Daigaku 2:167–188, 1935). Moreover, an efficient algorithm using MATLAB software to investigate the orders of starlikeness and convexity is presented as the first of a series. The given orders of starlikeness and convexity are then compared with some significant results in the literature to demonstrate the accuracy of our approach. Ultimately, the close-to-convexity of Hρ,r(z) as well as the lemniscate convexity have been evaluated using mathematical lemmas that have been given in Fejér (1936), Madaan et al. (Filomat 33(7):1937–1955, 2019) and Ozaki (1935). Further work regarding the function Hρ,r(z) is underway and can be presented in forthcoming articles.