[Paper Review] Lemniscate Convexity and Other Properties of Generalized Bessel Functions

Vibha Madaan, Ajay Kumar+1 more|arXiv (Cornell University)|Feb 12, 2019

Analytic and geometric function theory6 references4 citations

TL;DR

This paper investigates lemniscate convexity and other geometric properties of generalized and normalized Bessel functions using differential subordination. It establishes sufficient conditions on parameters $p$, $b$, and $c$ such that the function $u_{p,b,c}(z)$ satisfies subordination to $\sqrt{1+z}$, implying lemniscate convexity or starlikeness, and derives analogous results for the normalized Lommel function $h_{\mu,p}(z)$, including Alexander transform properties.

ABSTRACT

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

Motivation & Objective

• To determine sufficient conditions on parameters $p$, $b$, and $c$ such that the generalized and normalized Bessel function $u_{p,b,c}(z)$ is lemniscate convex or lemniscate starlike.
• To establish subordination conditions for $-(4(p+(b+1)/2)/c)u'_{p}(z) \prec \sqrt{1+z}$.
• To analyze the geometric properties of the normalized Lommel function $h_{\mu,p}(z)$, particularly its subordination to $\sqrt{1+z}$.
• To investigate the Alexander transform of $h_{\mu,p}(z)$ and its implications for geometric function classes.

Proposed method

• The study employs differential subordination theory, particularly the method of Miller and Mocanu, to analyze subordination conditions for analytic functions in the unit disk.
• The generalized Bessel function $u_{p,b,c}(z)$ is defined via a hypergeometric-type series involving the Pochhammer symbol and is shown to be entire.
• Key differential equations are derived for $u_{p,b,c}(z)$ and $h_{\mu,p}(z)$, which are used to establish subordination via functional inequalities.
• The proofs rely on constructing auxiliary functions $p(z)$ and applying Lemma 3.1 to verify non-vanishing of a complex functional $\psi(r,s,t;z)$ under given parameter constraints.
• The analysis involves bounding real parts and moduli of complex expressions involving $\theta$, $m$, and $z$ in the unit disk to ensure $\psi \neq 0$, thus confirming subordination.
• Parameter constraints are derived by ensuring the real part of a complex expression remains bounded away from zero under angular and modulus constraints.

Experimental results

Research questions

• RQ1 Under what conditions on $p$, $b$, and $c$ is the generalized Bessel function $u_{p,b,c}(z)$ lemniscate convex?
• RQ2 When does $-(4(p+(b+1)/2)/c)u'_{p}(z)$ subordinate to $\sqrt{1+z}$?
• RQ3 For which $\mu$ and $p$ does the normalized Lommel function $h_{\mu,p}(z)$ satisfy $1 + (zh''_{\mu,p}(z)/h'_{\mu,p}(z)) \prec \sqrt{1+z}$?
• RQ4 What geometric properties does the Alexander transform of $h_{\mu,p}(z)$ exhibit?

Key findings

• Sufficient conditions on $p$, $b$, and $c$ are derived such that $|(1 + zu''_p(z)/u'_p(z))^2 - 1| < 1$, ensuring lemniscate convexity of $u_p(z)$.
• The condition $-(4(p+(b+1)/2)/c)u'_p(z) \prec \sqrt{1+z}$ holds when the parameters satisfy specific inequalities involving $p$, $b$, and $c$.
• For the normalized Lommel function $h_{\mu,p}(z)$, the condition $1 + (zh''_{\mu,p}(z)/h'_{\mu,p}(z)) \prec \sqrt{1+z}$ is satisfied when $\operatorname{Re} \mu > -1$ and $p$ satisfies $\frac{3\operatorname{Re} \mu}{2\sqrt{2}} - \sqrt{3}\left|\frac{(\mu+1)^2 - p^2}{4} - 2\mu - 2\right| > \frac{13\sqrt{3}}{4} - \frac{15}{8\sqrt{2}} + \frac{1}{4}$.
• The Alexander transform of $h_{\mu,p}(z)$ is shown to preserve subordination to $\sqrt{1+z}$ under the derived parameter constraints.
• The function $u_p(z)$ is entire and satisfies the differential equation $4z^2 u''_p(z) + 4\kappa z u'_p(z) + c z u_p(z) = 0$, where $\kappa = p + (b+1)/2$.