Normalized generalized Bessel function and its geometric properties

Abstract

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