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Starlikeness of a generalized Bessel function. (English) Zbl 1409.33003

Summary: This paper investigates three functions f\({}_{a,v}\), g\({}_{a,v}\) and h\({}_{a,v}\) in the class \(\mathcal{A}\) consisting of analytic functions \(f\) in the unit disk satisfying \(f(0) = f^\prime(0) -1 =0\). Here \(a\in\{1,2,3,\dots\}\), and \(v\) is real. Each function is related to the generalized Bessel function. The radius of starlikeness of positive order is obtained for each of the three functions. Further, the best range on \(v\) is determined for a fixed \(a\) to ensure the functions f\({}_{a,v}\) and g\({}_{a,v}\) are starlike of positive order in the entire unit disk. When \(a=1\), the results obtained reduced to earlier known results.

MSC:

33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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Full Text: arXiv Euclid