Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$. Let $\mathcal{H}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$, and $\mathcal{H}_0 (z_0,w_0) = \{ f \in \mathcal{H}_0 : f(z_0) =w_0\}$. In this paper, we explicitly determine the variability region of $f''(z_0)$ when $f$ ranges over $\mathcal{H}_0 (z_0,w_0)$. We also show a geometric view of our main result by Mathematica.
Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$, and $\mathcal{H}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$. In this paper, we establish the third order Dieudonné Lemma, and apply it to explicitly determine the variability region $\{f'''(z_0): f\in \mathcal{H}_0,f(z_0) =w_0, f'(z_0)=w_1\}$ for given $z_0,w_0,w_1$ and give the form of all the extremal functions.
Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$, and $\mathcal{H}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$. In this paper, we establish the fourth-order Dieudonné's Lemma and apply it to determine the variability region $\{f^{(4)}(z_0): f\in \mathcal{H}_0,f(z_0) =w_0, f'(z_0)=w_1, f''(z_0)=w_2\}$ for given $z_0,w_0,w_1,w_2$ and give the form of all the extremal functions.
In this article we consider a family $\mathcal{C}(A, B)$ of analytic and locally univalent functions on the open unit disc $\ID=\{z :|z|<1\}$ in the complex plane that properly contains the well-known Janowski class of convex univalent functions. In this article, we determine the exact set of variability of $\log(f'(z_0))$ with fixed $z_0 \in \ID$ and $f"(0)$ whenever $f$ varies over the class $\mathcal{C}(A, B)$.
In this short note we employ the Briot-Bouquet differential subordination to determine the best possible inclusion relation within a certain family of analytic functions defined by the Salagean derivative.
In this paper, we obtain sharp bounds for the norm of pre--Schwarzian derivatives of certain analytic functions. Initially this problem was handled by H. Rahmatan, Sh. Najafzadeh and A. Ebadian [Stud Univ Babeş--Bolyai Math {\bf61}(2): 155--162, 2016]. We pointed out that the proofs by Rahmatan et al. are incorrect and present correct proofs.
Let the function $φ$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $φ(\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $φ$, $$|D_φ(z)|:=\frac{(1-|z|^2)|φ'(z)|}{1-|φ(z)|^2}.$$ In particular, we show how the Schwarzian derivative of $φ$ reveals the nature of the critical points.
In this paper, by making use of a certain family of fractional derivative operators in the complex domain, we introduce and investigate a new subclass $\mathcal{P}_{τ,μ}(k,δ,γ)$ of analytic and univalent functions in the open unit disk $\mathbb{U}$. In particular, for functions in the class $\mathcal{P}_{τ,μ}(k,δ,γ)$, we derive sufficient coefficient inequalities, distortion theorems involving the above-mentioned fractional derivative operators, and the radii of starlikeness and convexity. In addition, some applications of functions in the class $\mathcal{P}_{τ,μ}(k,δ,γ)$ are also pointed out.
We continue our study on variability regions in \cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region of variability $V_Ω^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g({\mathbb D}) \subset Ω, \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$ for each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 , \ldots , c_n) \in \mathbb{C}^{n+1}$, when $Ω\subsetneq\mathbb{C}$ is a convex domain, and $P$ is a conformal map of the unit disk ${\mathbb D}$ onto $Ω$. In the present article, we first show that in the case $n=0$, $j=-1$ …
Relationship between the value-sharing behavior of an entire function and its derivative, and the analytic structure of a nonlinear differential equation
The representation for the sharp constant ${\rm K}_{n, p}$ in an estimate of the modulus of the $n$-th derivative of an analytic function in the upper half-plane ${\mathbb C}_+$ is considered. It is assumed that the boundary value of the real part of the function on $\partial{\mathbb C}_+$ belongs to $L^p$. The representation for ${\rm K}_{n, p}$ comprises an optimization problem by parameter inside of the integral. This problem is solved for $p=2(m+1)/(2m+1-n)$, $n\leq 2m+1$, and for some first derivatives of even order in the case $p=\infty$. The formula for ${\rm K}_{n,\; 2(m+1)/(2m+1-n)}$ …
Let $n$ be a positive integer. Let $\mathbf U$ be the unit disk, $p\ge 1$ and let $h^p(\mathbf U)$ be the Hardy space of harmonic functions. Kresin and Maz'ya in a recent paper found the representation for the function $H_{n,p}(z)$ in the inequality $$|f^{(n)} (z)|\leq H_{n,p}(z)|\Re(f-\mathcal P_l)|_{h^p(\mathbf U)}, \Re f\in h^p(\mathbf U), z\in \mathbf U,$$ where $\mathcal P_l$ is a polynomial of degree $l\le n-1$. We find or represent the sharp constant $C_{p,n}$ in the inequality $H_{n,p}(z)\le \frac{C_{p,n}}{(1-|z|^2)^{1/p+n}}$. This extends a recent result of the second author and Marko…
Let $Ω$ be a convex domain in the complex plane ${\mathbb C}$ with $Ω\not= {\mathbb C}$, and $P$ be a conformal map of the unit disk ${\mathbb D}$ onto $Ω$. Let ${\mathcal F}_Ω$ be the class of analytic functions $g$ in ${\mathbb D}$ with $g({\mathbb D}) \subset Ω$, and $H_1^\infty ({\mathbb D})$ be the closed unit ball of the Banach space $H^\infty ({\mathbb D})$ of bounded analytic functions $ω$ in ${\mathbb D}$, with norm $\| ω\|_\infty = \sup_{z \in {\mathbb D}} |ω(z)|$. Let ${\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text{there exists} \; ω\in H_1^\infty ({\mat…
A proof is reconstructed for a useful theorem on the zeros of derivatives of analytic functions due to H. M. Macdonald, which appears to be now little known. The Theorem states that, if a function $f(z)$ is analytic inside a bounded region bounded by a contour on which the modulus of $f(z)$ is constant, then the number of zeros (counted according to multiplicity) of $f(z)$ and of its derivative in the region differ by unity.
We review the relation between the classical formulas of the pre-Schwarzian and Schwarzian derivatives of locally univalent analytic functions and the derivatives of the generating functions of the methods due to Newton and Halley, respectively, for approximating zeros. We extend these relations to the cases when the functions considered are harmonic.
The theory of $q$-analogs frequently occurs in a number of areas, including the fractals and dynamical systems. The $q$-derivatives and $q$-integrals play a prominent role in the study of $q$-deformed quantum mechanical simple harmonic oscillator. In this paper, we define a symmetric $q$-derivative operator and study new family of univalent functions defined by use of that operator. We establish some new relations between functions satisfying analytic conditions related to conical sections.
We establish a criterion for local boundedness and hence normality of a family $\F$ of analytic functions on a domain $D$ in the complex plane whose corresponding family of derivatives is locally bounded. Furthermore we investigate the relation between domains of normality of a family $\F$ of meromorphic functions and its corresponding Schwarzian derivative family. We also establish some criterion for the Schwarzian derivative family of a family $\F$ of analytic functions on a domain $D$ in the complex plane to be a normal family.
We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $α$, with $0<α<1$, in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We show that whenever such a bounded point derivation exists at a boundary point $b$, it may be evaluated by taking a limit of…
Analytic summation of series involving higher-order derivatives of Chebyshev polynomials of the second kind and their applications to convolved linear recurrent sequences
We introduce a join construction as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom's A_f condition.
In this paper, we continue the study of the polar analytic functions, a notion introduced in \cite{BBMS1} and successfully applied in Mellin analysis. Here we obtain another version of the Cauchy integral formula and a residue theorem for polar Mellin derivatives, employing the new notion of logarithmic pole. The identity theorem for polar analytic functions is also derived. As applications we obtain an analogue of Boas' differentiation formula for polar Mellin derivatives, and an extension of the classical Bernstein inequality to polar Mellin derivatives. Finally we give an analogue of the we…
