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Abstract
In this note our aim is to determine the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization. The key tool in the proof of our main result is the Mittag-Leffler expansion for Bessel functions of the first kind and the fact that, according to Ismail and Muldoon [IM2], the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind.
Results & Lemmas (2)
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Theorem 1.
Theorem 1. Let 1 > β ≥0. Then the following assertions are true: a. If ν ∈(−1, 0), then r∗ β (fν) = xν,β, where xν,β denotes the unique…
Theorem 1. Let 1 > β ≥0. Then the following assertions are true: a. If ν ∈(−1, 0), then r∗ β (fν) = xν,β, where xν,β denotes the unique positive root of the equation zI′ ν(z)−βνIν(z) = 0. Moreover, if ν > 0, then we have r∗ β (fν) = xν,β,1, where xν,β,1 is the smallest positive root of the equation zJ′ ν(z) −βνJν(z) = 0. b. If ν > −1, then r∗ β (gν) = yν,β,1, where yν,β,1 is the smallest positive root of the equation zJ′ ν(z) + (1 −β −ν)Jν(z) = 0. c. If ν > −1, then r∗ β (hν) = zν,β,1, where zν,
Corollary 1. · radius
Corollary 1. The following assertions are true: a. If ν ∈(−1, 0), then the radius of starlikeness of fν is xν,0, where xν,0 is the unique…
Corollary 1. The following assertions are true: a. If ν ∈(−1, 0), then the radius of starlikeness of fν is xν,0, where xν,0 is the unique positive root of the equation I′ ν(z) = 0. If ν > 0, then the radius of starlikeness of the function fν is xν,0,1, which denotes the smallest positive root of the equation J′ ν(z) = 0. b. If ν > −1, then the radius of starlikeness of the function gν is yν,0,1, which denotes the smallest positive root of the equation zJ′ ν(z)+ (1 −ν)Jν(z) = 0. c. If ν > −1, the
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