Oqlah Al-Refai

For $n\ge 3$, we study the subclass ${\mathcal{C}}_n$ of univalent functions f with $a_2=\dots =a_n=0$ in which the whole allowance of the Al-Refai–Darus criterion is assigned to the nth derivative of $z/f$. The map $f\mapsto {(z/f)}^{\left(n\right)}$ is a bijection of ${\mathcal{C}}_n$ onto a closed ball of $H^{\infty }$, which turns extremal problems into problems for bounded analytic functions. We prove sharp growth, distortion, covering-radius and coefficient bounds with the $(n+j)$-fold symmetric extremal functions $z/(1+c{z}^{n+j})$; the exact region of $(a_{n+1},a_{n+2})$, whose boundary is attained by Möbius parameters written in closed form; an exact radius for dilations of products $f_1{f}_2/z$, sharp both for Aksentev’s inequality and for univalence. The vanishing of $a_2,\dots ,a_n$ does not by itself imply n-fold symmetry; the n-fold symmetric members of ${\mathcal{C}}_n$ correspond exactly to $g\left(z^n\right)$ with g in the same ball, which provides the exact region of $(a_{n+1},a_{2n+1})$. For the larger class defined by $a_2=0$ and the criterion, we obtain sharp bounds as well. The comparison with n-fold symmetric univalent functions records a gap; it is not a sharpening of known results.

The Al-Refai--Darus criterion guarantees univalence of a normalized function \( f \) when a weighted sum of \( |(z/f)^{(k)}(0)| \), \( 2\le k\le n-1 \), and of \( \sup|(z/f)^{(n)}| \) does not exceed \( 1 \). For \( n\ge3 \) we study the subclass \( \mathcal C_n \) with \( a_2=\dots=a_n=0 \), in which the whole allowance goes to the \( n \) th derivative. The map \( f\mapsto(z/f)^{(n)} \) is a bijection of \( \mathcal C_n \) onto a closed ball of \( H^\infty \), which turns extremal problems into problems for bounded analytic functions. We prove sharp growth, distortion, covering-radius and coefficient bounds with the \( (n+j) \)-fold symmetric extremal functions \( z/(1+cz^{n+j}) \); the exact region of \( (a_{n+1},a_{n+2}) \), whose boundary is attained by M\"obius parameters written in closed form; and an exact radius for dilations of products \( f_1f_2/z \), sharp both for Aksentev's inequality and for univalence. The \( n \)-fold symmetric members of \( \mathcal C_n \) correspond to \( g(z^n) \) with \( g \) in the same ball, which gives the exact region of \( (a_{n+1},a_{2n+1}) \). For the larger class defined by \( a_2=0 \) and the criterion we obtain sharp bounds as well. The comparison with \( n \)-fold symmetric univalent functions records a gap; it is not a sharpening of known results.

Let $\mathcal{A}$ denote the class of functions $g\left(z\right)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k$ analytic and normalized in $\mathbb{U}=\{z:|z|<1\}$, and let $D=\{g\in \mathcal{A}:|g^{″}\left(z\right)|\le 1,z\in \mathbb{U}\}$. We give a short, self-contained proof that D consists entirely of univalent functions by a direct line-integral estimate combined with the classical Noshiro–Warschawski theorem. We then solve, in closed form and with explicit extremal functions, two classical extremal coefficient problems for D: the Fekete–Szegő problem and the second Hankel determinant, using the exact Schur parametrization of bounded analytic functions applied to $g^{″}$. We also establish a sharp bound on a single coefficient $|a_n|$ for every $n\ge 2$, obtained by an elementary Parseval argument that does not require the full joint Schur parametrization needed for the Fekete–Szegő and Hankel problems. We then compose D with the Owa–Srivastava fractional derivative operator ${\mathsf{\Omega }}_z^{\delta }$ ($\delta \in [0,1)$), used to build fractional bi-univalent subclasses, to define the one-parameter family $D\left(\delta \right)=\{g\in \mathcal{A}:|{\left({\mathsf{\Omega }}_z^{\delta }g\right)}^{″}\left(z\right)|\le 1\}$, and extend all three results to $D\left(\delta \right)$, again with explicit extremal functions. Because the underlying weights grow at different rates in the fractional parameter $\delta$, the Fekete–Szegő problem exhibits a genuine, monotone bifurcation as $\delta$ ranges over $[0,1)$, while the Hankel and general-coefficient bounds decrease monotonically—effects that are structural consequences of the fractional operator rather than a simple rescaling of the classical case. All results are verified in two independent numerical ways: the Schur parametrization lemma itself is checked against an explicit, independently constructed rational Schur function whose Taylor coefficients are extracted numerically via the Cauchy integral formula, and the sharp bounds are checked by Monte Carlo maximization over the admissible Schur parameters. Both checks are explained in detail and agree with the closed forms to at least four decimal places. Unlike the Carathéodory class of functions with positive real part, whose coefficient body is fixed at the origin (value 1) and is classically parametrized without an extra recursive step, the Schur class $\mathcal{B}$ used here does not pin down $\phi \left(0\right)$, so its exact coefficient body genuinely requires the two-step recursive Schur parametrization applied below to $g^{″}$; identifying and exploiting this distinction is part of the technical contribution of the paper.

In this paper, specific $ L^p $ estimates for generalized Marcinkiewicz operators correlated to surfaces of revolution are proved. These estimates and the extrapolation procedure of Yano are employed to confirm the $ L^p $ boundedness of the above-mentioned integrals under weaker assumptions on the singular kernels. Our findings generalize and improve several known results.

In this article, certain sharp $L^p$ estimates for a specific class of generalized Marcinkiewicz operators with mixed homogeneity associated to surfaces of revolution are established. By virtue of Yano’s extrapolation argument, beside these estimates, the $L^p$ boundedness of the aforementioned operators under weaker assumptions on the kernels is confirmed. The obtained results in this article are fundamental extensions and improvements of numerous previously known results on parabolic generalized Marcinkiewicz integrals.