Lemniscate Convexity and Other Properties of Generalized Bessel Functions

Abstract

Sufficient conditions on associated parameters $p,b$ and $c$ are obtained so that the generalized and \textquotedblleft{normalized}\textquotedblright{} Bessel function $u_p(z)=u_{p,b,c}(z)$ satisfies $|(1+(z{u}_p^{\prime \prime }(z)/u_p^{\prime }(z)){)}^2-1|<1$ or $|((z{u}_p(z){)}^{\prime }/u_p(z){)}^2-1|<1$. We also determine the condition on these parameters so that $-(4(p+(b+1)/2)/c)u_p^{\prime }(z)\prec \sqrt{1+z}$. Relations between the parameters $\mu$ and $p$ are obtained such that the normalized Lommel function of first kind $h_{\mu ,p}(z)$ satisfies the subordination $1+(z{h}_{\mu ,p}^{\prime \prime }(z)/h_{\mu ,p}^{\prime }(z))\prec \sqrt{1+z}$. Moreover, the properties of Alexander transform of the function $h_{\mu ,p}(z)$ are discussed.

Cited by in corpus (1)

• Radii of Starlikeness and Convexity of Bessel Functions