Details

Autor(en) / Beteiligte

• Bulboacă, Teodor
• Zayed, Hanaa M.

Titel

Analytical and geometrical approach to the generalized Bessel function

Ist Teil von

• Journal of inequalities and applications, 2024-12, Vol.2024 (1), p.51-17

Ort / Verlag

Cham: Springer International Publishing

Erscheinungsjahr

2024

Link zum Volltext

Quelle

SpringerLink (Online service)

Beschreibungen/Notizen

• 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 , for ρ , r ∈ C ∗ : = C ∖ { 0 } . Precisely, we will use a sharp estimate for the Pochhammer symbol, that is, Γ ( a + n ) / Γ ( a + 1 ) > ( a + α ) n − 1 , or equivalently ( a ) n > a ( a + α ) n − 1 , that was firstly proved by Baricz and Ponnusamy for n ∈ N ∖ { 1 , 2 } , a > 0 and α ∈ [ 0 , 1.302775637 … ] 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 } , a > 0 and 0 ≤ α ≤ 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.