Radii of starlikeness and convexity of generalized $k-$Besse

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Abstract

The main purpose of this paper is to determine the radii of starlikeness and convexity of the generalized $\emph{k}-$Bessel functions for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-Pólya class plays an crucial role in this paper. Moreover, the interlacing properties of the zeros of $\emph{k}-$Bessel function and it

Results & Lemmas (5)

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Lemma 1.1. Lemma 1.1. Let k > 0, c > 0 and ν > 0. Then the function z 7→kWν,c(z) has inﬁnitely many zeros which are all real. Denoting by kων,c,n the…

Lemma 1.1. Let k > 0, c > 0 and ν > 0. Then the function z 7→kWν,c(z) has inﬁnitely many zeros which are all real. Denoting by kων,c,n the nth positive zero of kWν,c(z), under the same conditions the Weierstrassian decomposition (1.6) kWν,c(z) = z 2 ν k Γk(ν + k) Y n≥1 1 − z2

Theorem 1.1. · radius Theorem 1.1. Let k > 0, c > 0, ν > 0 and α ∈[0, 1). Then the following assertions are true. a. The radius of starlikeness of order α of the…

Theorem 1.1. Let k > 0, c > 0, ν > 0 and α ∈[0, 1). Then the following assertions are true. a. The radius of starlikeness of order α of the function kfν,c is the smallest positive root of the equation krkW ′ ν,c(r) −ναkWν,c(r) = 0. b. The radius of starlikeness of order α of the function kgν,c is the smallest positive root of the equation rkW ′ ν,c(r) − α + ν k −1 kWν,c(r) = 0. c. The radius of starlikeness of order α of the function khν,c is the smallest positive root of

Theorem 1.2. Theorem 1.2. Let k > 0, c > 0 and ν > 0.

Theorem 1.2. Let k > 0, c > 0 and ν > 0.

Theorem 1.3. · radius Theorem 1.3. Let k > 0, c > 0, ν > 0 and α ∈[0, 1). a. The radius of convexity of order α of the function kfν,c is the smallest root of the…

Theorem 1.3. Let k > 0, c > 0, ν > 0 and α ∈[0, 1). a. The radius of convexity of order α of the function kfν,c is the smallest root of the equation 1 + r kW ′′ ν,c(r) kW ′ν,c(r) + k ν −1 r kW ′ ν,c(r) kWν,c(r) = α. b. The radius of convexity of order α of the function kgν,c is the smallest root of the equation 1 + r

Theorem 1.4. · radius Theorem 1.4. Let k > 0, c > 0, ν > 0. a. The radius of convexity rc(kgν,c) of the function z 7→kgν,c(z) = 2 ν k Γk(ν + k)z1−ν k kWν,c(z),

Theorem 1.4. Let k > 0, c > 0, ν > 0. a. The radius of convexity rc(kgν,c) of the function z 7→kgν,c(z) = 2 ν k Γk(ν + k)z1−ν k kWν,c(z),

Function classes studied: