[Paper Review] Spirallikeness of shifted hypergeometric functions

Toshiyuki Sugawa, Limei Wang|arXiv (Cornell University)|Apr 18, 2016

Analytic and geometric function theory10 references3 citations

TL;DR

This paper investigates the spirallikeness of shifted hypergeometric functions $ f(z) = z{}_2F_1(a,b;c;z) $ with complex parameters $ a, b, c $, establishing necessary and sufficient conditions for $ f $ to be $ \lambda $-spirallike for $ \lambda \in (-\pi/2, \pi/2) $. The key contribution is a new sufficient condition based on the real parts of parameter combinations, yielding explicit examples of spirallike but not starlike functions, including cases with complex parameters.

ABSTRACT

In the present paper, we study spirallikenss (including starlikeness) of the shifted hypergeometric function $f(z)=z_2F_1(a,b;c;z)$ with complex parameters $a,b,c,$ where $_2F_1(a,b;c;z)$ stands for the Gaussian hypergeometric function. First, we observe the asymptotic behaviour of $_2F_1(a,b;c;z)$ around the point $z=1$ to obtain necessary conditions for $f$ to be $λ$-spirallike for a given $λ$ with $- π/2< λ

Motivation & Objective

• To extend geometric function theory to shifted hypergeometric functions with complex parameters, particularly beyond the known real-parameter cases.
• To address the lack of results on spirallikeness in hypergeometric functions, as prior work focused mainly on starlikeness and convexity for real parameters.
• To derive necessary and sufficient conditions for $ f(z) = z{}_2F_1(a,b;c;z) $ to be $ \lambda $-spirallike, generalizing starlikeness.
• To construct explicit examples of $ \lambda $-spirallike functions that are not starlike, demonstrating the broader scope of the new conditions.

Proposed method

• Analyzing the asymptotic behavior of $ {}_2F_1(a,b;c;z) $ near $ z=1 $ to derive necessary conditions for $ \lambda $-spirallikeness.
• Applying a criterion based on the real part of $ e^{-i\lambda} \frac{zf'(z)}{f(z)} $, which characterizes $ \lambda $-spirallikeness via a real part condition.
• Deriving a sufficient condition using the parameters' real and imaginary parts, particularly through the discriminant $ LN - M^2 \geq 0 $, where $ L, M, N $ are real parts of symmetric combinations of $ a, b, c $.
• Using the Alexander relation $ zg'(z) = f(z) $ to relate starlikeness of $ f $ to convexity of $ g $, enabling transfer of results between classes.
• Applying the condition $ \sigma(f) \geq \alpha $ for strong starlikeness and relating it to $ \lambda $-spirallikeness via $ \alpha = 1 - \frac{2}{\pi}|\lambda| $.
• Constructing explicit examples using parameter constraints such as $ a + b = s \in \mathbb{R} $, $ ab = qe^{i\lambda} $, and deriving bounds on $ q $ and $ s $ for $ \lambda $-spirallikeness.

Experimental results

Research questions

• RQ1 What are the necessary conditions for the shifted hypergeometric function $ f(z) = z{}_2F_1(a,b;c;z) $ to be $ \lambda $-spirallike for complex parameters $ a, b, c $?
• RQ2 What sufficient conditions ensure $ f(z) $ is $ \lambda $-spirallike, especially when $ f $ is not starlike?
• RQ3 Can explicit examples of $ \lambda $-spirallike but not starlike functions be constructed for complex parameters?
• RQ4 How do parameter symmetries and real parts of $ ab $, $ b+c $, and $ c-b $ influence spirallikeness?
• RQ5 What is the role of the discriminant $ LN - M^2 \geq 0 $ in determining the spirallikeness of $ f(z) $?

Key findings

• For $ f(z) = z{}_2F_1(2, b+is; c+is; z) $ with $ 3 \leq b+c $ and $ b \leq c $, the function is starlike, and $ g(z) = z{}_2F_1(1, b+is; c+is; z) $ is convex.
• The function $ z{}_2F_1(2, b; 3 - \bar{b}; z) $ satisfies all conditions of Theorem 1.2 with equality in $ \operatorname{Re}[ab] = p $, but is not starlike if $ \operatorname{Re} b > 3/2 $, showing the sharpness of the inequality condition.
• For $ a = 2e^{i\lambda}\cos\lambda $, $ b > 0 $, the function $ z{}_2F_1(a, b; a + b + 1; z) $ is $ \lambda $-spirallike for all $ b > 0 $, with $ L > 0 $, $ M = 0 $, $ N > 0 $, ensuring $ LN - M^2 > 0 $.
• Corollary 4.5 provides a sufficient condition for $ \lambda $-spirallikeness when $ a + b = s \in \mathbb{R} $, $ ab = qe^{i\lambda} $, requiring $ 2s - \frac{q}{\cos\lambda} \geq \frac{4\cos^2\lambda - 1}{2\cos^2\lambda + 1} $.
• An example with $ \lambda = \pi/4 $, $ s = 35/8 $, $ q = 125/(16\sqrt{2}) $, satisfies the inequality and yields a $ \lambda $-spirallike function that is not starlike, as $ \operatorname{Re}(zf'(z)/f(z)) \approx -0.0374 $ at $ z = e^{i\pi/4} $.
• The function $ z{}_2F_1(1, \gamma+1; \gamma+2; z) $ is convex for $ \operatorname{Re} \gamma \geq 0 $, consistent with known results in Ruscheweyh (1991).