[Paper Review] Radii of convexity of integral operators

Parvaneh Najmadi, Sh. Najafzadeh+1 more|arXiv (Cornell University)|Apr 11, 2018

Analytic and geometric function theory6 references5 citations

TL;DR

This paper investigates the radii of convexity for two integral operators involving products of derivatives and normalized analytic functions. By applying subordination principles and inequalities from linear invariant families, the authors derive explicit formulas for the radius of convexity under specific coefficient and order constraints, providing sharp bounds for univalence and convexity in the unit disk.

ABSTRACT

The object of the present paper is to study of radius of convexity two certain integral operators as follows \begin{equation*} F(z):=\int_{0}^{z}\prod_{i=1}^{n}\left(f'_i(t) ight)^{γ_i}{ m d}t \end{equation*} and \begin{equation*} J(z):=\int_{0}^{z}\prod_{i=1}^{n}\left(f'_i(t) ight)^{γ_i}\prod_{j=1}^{m} \left(\frac{g_j(z)}{z} ight)^{λ_j}{ m d}t, \end{equation*} where $γ_i, λ_i\in\mathbb{C}$, $f_i$ $(1\leq i\leq n)$ and $g_j$ $(1\leq j\leq m)$ belong to the certain subclass of analytic functions.

Motivation & Objective

• To determine the radius of convexity for a generalized integral operator involving products of analytic functions and their derivatives.
• To extend existing results on starlikeness and convexity to more general integral operators with complex exponents.
• To establish sufficient conditions under which the integral operator remains convex in a disk of radius r < 1.
• To generalize prior work on integral operators by incorporating multiple functions and complex exponents in the kernel.
• To provide explicit, closed-form expressions for the radius of convexity based on the orders of the component functions and the norms of the exponents.

Proposed method

• The authors define two integral operators: $ F(z) = \int_0^z \prod_{i=1}^n (f_i'(t))^{\gamma_i} dt $ and $ J(z) = \int_0^z \prod_{i=1}^n (f_i'(t))^{\gamma_i} \prod_{j=1}^m \left(\frac{g_j(z)}{z}\right)^{\lambda_j} dt $, where $ f_i, g_j $ are analytic in the unit disk.
• They use the condition $ \mathfrak{Re}\left(1 + \frac{zJ''(z)}{J'(z)}\right) \geq 0 $ to characterize convexity, reducing the problem to estimating the real part of a sum involving $ \frac{zf_i''(z)}{f_i'(z)} $ and $ \frac{zg_j'(z)}{g_j(z)} - 1 $.
• For functions in $ \mathcal{U}_{\alpha_i} $, the authors apply Lemma 1.1 to bound $ \left| \frac{zf_i''(z)}{f_i'(z)} \right| $ in terms of $ \alpha_i $ and $ |z| $, leading to a quadratic lower bound in $ r = |z| $.
• For $ g_j \in \mathcal{S}^*(\xi_j) $, they use subordination to $ \frac{1 + (1 - 2\xi_j)z}{1 - z} $, which yields a bound on $ \mathfrak{Re}\left( \frac{zg_j'(z)}{g_j(z)} - 1 \right) $ in terms of $ \xi_j $ and $ r $.
• By summing over all components and using $ \sum |\gamma_i| \leq M $, $ \sum |\lambda_j| \leq N $, they derive a rational function $ \varphi(r) $ whose positivity determines the radius of convexity.
• The radius $ r_c(M,N) $ is obtained as the smallest positive root of the quadratic equation derived from setting the numerator of $ \varphi(r) $ to zero, yielding a closed-form expression.

Experimental results

Research questions

• RQ1 What is the largest disk $ |z| < r $ within the unit disk where the integral operator $ J(z) $ is convex?
• RQ2 How do the orders of the component functions $ f_i $ and $ g_j $, and the norms of the exponents $ \gamma_i, \lambda_j $, affect the radius of convexity?
• RQ3 Can a unified formula be derived for the radius of convexity when the functions belong to generalized classes like $ \mathcal{U}_{\alpha_i} $ and $ \mathcal{S}^*(\xi_j) $?
• RQ4 Under what conditions is the integral operator $ J(z) $ guaranteed to be convex in a disk centered at the origin?
• RQ5 What is the sharpness of the derived radius formula, and can it be reduced to known cases like convex or starlike functions?

Key findings

• The radius of convexity for the integral operator $ J(z) $ is given by the explicit formula $ r_c(M,N) = \frac{\sqrt{[(\xi-1)N - \alpha M]^2 - 2[(\xi-1)N - M - 1]} - (\xi-1)N + \alpha M}{2[(\xi-1)N - M - 1]} $, where $ \alpha = \max\{\alpha_1, \dots, \alpha_n\} $, $ \xi = \max\{\xi_1, \dots, \xi_m\} $, $ M = \sum |\gamma_i| $, and $ N = \sum |\lambda_j| $.
• When $ \alpha = 1 $, the formula simplifies to $ r_c(M,N) = \frac{\sqrt{[(\xi-1)N - M]^2 - 2[(\xi-1)N - M - 1]} - (\xi-1)N + M}{2[(\xi-1)N - M - 1]} $, corresponding to the case where all $ f_i $ are convex functions.
• For locally convex univalent functions of order $ \beta_i $, the radius of convexity is the positive root of the quadratic equation $ -[2(1-\xi)N + \beta M + 1]r^2 - [2(1-\xi)N + \beta M]r + 1 = 0 $, with $ \beta = \max\{\beta_1, \dots, \beta_n\} $.
• The derived bounds are sharp, as equality is achieved for the function $ \frac{z}{(1-z)^{2(1-\xi)}} $, which realizes the extremal behavior in the subordination estimate.
• The method relies on combining subordination principles with inequalities from linear invariant families, particularly Lemma 1.1, to control the real part of the logarithmic derivative.
• The results generalize previous findings on integral operators by incorporating complex exponents and multiple functions, offering a unified framework for convexity radii.