Analytical and geometrical approach to the generalized Bessel function

Abstract

Abstract In continuation of Zayed and Bulboacă work in (J. Inequal. Appl. 2022:158, 2022), this paper discusses the geometric characterization of the normalized form of the generalized Bessel function defined by V ρ , r ( z ) : = z + ∑ k = 1 ∞ ( − r ) k 4 k ( 1 ) k ( ρ ) k z k + 1 , z ∈ U , $$\begin{aligned} \mathrm{V}_{\rho,r}(z):=z+\sum_{k=1}^{\infty} \frac{(-r)^{k}}{4^{k}(1)_{k}(\rho )_{k}}z^{k+1}, \quad z\in \mathbb{U}, \end{aligned}$$ for ρ , r ∈ C ∗ : = C ∖ { 0 } $\rho, r\in \mathbb{C}^{\ast}:=\mathbb{C}\setminus \{0\}$ . Precisely, we will use a sharp estimate for the Pochhammer symbol, that is, Γ ( a + n ) / Γ ( a + 1 ) > ( a + α ) n − 1 $\Gamma (a+n)/\Gamma (a+1)>(a+\alpha )^{n-1}$ , or equivalently ( a ) n > a ( a + α ) n − 1 $(a)_{n}>a(a+\alpha )^{n-1}$ , that was firstly proved by Baricz and Ponnusamy for n ∈ N ∖ { 1 , 2 } $n\in \mathbb{N}\setminus \{1,2\}$ , a > 0 $a>0$ and α ∈ [ 0 , 1.302775637 … ] $\alpha \in [0,1.302775637\ldots ]$ in (Integral Transforms Spec. Funct. 21(9):641–653, 2010), and then proved in our paper by another method to improve it using the partial derivatives and the two-variable functions’ extremum technique for n ∈ N ∖ { 1 , 2 } $n\in \mathbb{N}\setminus \{1,2\}$ , a > 0 $a>0$ and 0 ≤ α ≤ 2 $0\leq \alpha \leq \sqrt{2}$ , and used to investigate the orders of starlikeness and convexity. We provide the reader with some examples to illustrate the efficiency of our theory. Our results improve, complement, and generalize some well-known (nonsharp) estimates, as seen in the Concluding Remarks and Outlook section.

Keywords

• Analytic
• Univalent
• Starlike
• Convex functions
• Convolution product
• Pochhammer symbol