Normalized generalized Bessel function and its geometric properties

Abstract The normalization of the generalized Bessel functions $\mathrm{U}_{\sigma,r}$ U σ , r $(\sigma,r\in \mathbb{C}\mathbbm{)}$ ( ∈ C ) defined by $$\begin{aligned} \mathrm{U}_{\sigma,r}(z)=z+\sum_{j=1}^{\infty} \frac{(-r)^{j}}{4^{j} (1)_{j}(\sigma )_{j}}z^{j+1} \end{aligned}$$ z = + ∑ j 1 ∞ − 4 was introduced, and some its geometric properties have been presented previously. main purpose present paper is to complete results given in literature employing a new procedure. We first used an identity for logarithmic gamma function as well inequality digamma establish sufficient conditions on parameters so that starlike or convex order α $(0\leq \alpha \leq 1)$ 0 ≤ α open unit disk. Moreover, starlikeness convexity considered where leading concept proofs comes from power series $f(z)=\sum_{j=1}^{\infty}A_{j}z^{j}$ f A classical Alexander theorem between classes functions. gave simple proof show our are not contradictory. Ultimately, close-to-convexity $(z\cos \sqrt{z} ) \ast \mathrm{U}_{\sigma,r}$ cos ∗ $(\sin z \frac {\mathrm{U}_{\sigma,r}(z^{2})}{z}$ sin 2 determined, “∗” stands convolution series.