Abstract
A normalized analytic function f is lemniscate starlike if the quantity \(zf'(z)/f(z)\) lies in the region bounded by the right half of the lemniscate of Bernoulli \(|w^2-1|=1\). It is Janowski starlike if the quantity \(zf'(z)/f(z)\) lies in the disk whose diametric end points are \((1-A)/(1-B)\) and \((1+A)/(1+B)\) for \(-1\le B<A\le 1\). The radii of lemniscate starlikeness and Janowski starlikeness have been determined for normalizations of q-Bessel functions, Bessel functions of first kind of order \(\nu \) and Lommel functions of first kind. Corresponding convexity radii are also determined.
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References
Aktaş, İ., Baricz, Á.: Bounds for radii of starlikeness of some \(q\)-Bessel functions. Results Math. 72(1–2), 947–963 (2017)
Ali, R.M., Jain, N.K., Ravichandran, V.: Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane. Appl. Math. Comput. 218(11), 6557–6565 (2012)
Baricz, Á.: Generalized Bessel Functions of the First Kind, Lecture Notes in Mathematics, vol. 1994. Springer-Verlag, Berlin (2010)
Baricz, Á., Dimitrov, D.K., Mező, I.: Radii of starlikeness and convexity of some \(q\)-Bessel functions. J. Math. Anal. Appl. 435(1), 968–985 (2016)
Baricz, Á., Dimitrov, D.K., Orhan, H., Yağmur, N.: Radii of starlikeness of some special functions. Proc. Am. Math. Soc. 144(8), 3355–3367 (2016)
Baricz, Á., Koumandos, S.: Turán-type inequalities for some Lommel functions of the first kind. Proc. Edinb. Math. Soc. 59(3), 569–579 (2016)
Baricz, Á., Kupán, P.A., Szász, R.: The radius of starlikeness of normalized Bessel functions of the first kind. Proc. Am. Math. Soc. 142(6), 2019–2025 (2014)
Baricz, Á., Szász, R.: The radius of convexity of normalized Bessel functions of the first kind. Anal. Appl. (Singap.) 12(5), 485–509 (2014)
Baricz, Á., Yağmur, N.: Geometric properties of some Lommel and Struve functions. Ramanujan J. 42(2), 325–346 (2017)
Bohra, N., Ravichandran, V.: Radii problems for normalized Bessel functions of first kind. Comput. Methods Funct. Theory 18(1), 99–123 (2018)
Gangadharan, A., Ravichandran, V., Shanmugam, T.N.: Radii of convexity and strong starlikeness for some classes of analytic functions. J. Math. Anal. Appl. 211(1), 301–313 (1997)
Goodman, A.W.: Univalent Functions, vol. I. Mariner Publishing Co. Inc., Tampa (1983)
Ismail, M.E.H.: The zeros of basic Bessel functions, the functions \(J_{\nu +ax}(x)\), and associated orthogonal polynomials. J. Math. Anal. Appl. 86(1), 1–19 (1982)
Janowski, W.: Extremal problems for a family of functions with positive real part and for some related families. Ann. Polon. Math. 23, 159–177 (1970/1971)
Koelink, H.T., Swarttouw, R.F.: On the zeros of the Hahn-Exton \(q\)-Bessel function and associated \(q\)-Lommel polynomials. J. Math. Anal. Appl. 186(3), 690–710 (1994)
Madaan, V., Kumar, A., Ravichandran, V.: Lemniscate convexity of generalized Bessel functions. Studia Sci. Math. Hungar. 56(4), 404–419 (2019)
Orhan, H., Aktaş, İ.: Bounds for radii of convexity of some \( q \)-Bessel functions (2017). arXiv:1702.04549
Sokól, J., Stankiewicz, J.: Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19, 101–105 (1996)
Verma, S., Ravichandran, V.: Radius problems for ratios of Janowski starlike functions with their derivatives. Bull. Malays. Math. Sci. Soc. 40(2), 819–840 (2017)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Reprint of the second (1944) edition. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1995)
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Communicated by See Keong Lee.
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The first author is supported by Senior Research Fellowship from University Grants Commission, New Delhi, Ref. No.: 1069/(CSIR-UGC NET DEC, 2016).
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Madaan, V., Kumar, A. & Ravichandran, V. Radii of Starlikeness and Convexity of Some Entire Functions. Bull. Malays. Math. Sci. Soc. 43, 4335–4359 (2020). https://doi.org/10.1007/s40840-020-00925-8
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DOI: https://doi.org/10.1007/s40840-020-00925-8