Coefficient inequality for transforms of parabolic starlike and uniformly convex functions

D. Vamshee Krishna; B. Venkateswarlu; T. RamReddy

Displaying similar documents to “Coefficient inequality for transforms of parabolic starlike and uniformly convex functions”

Coefficient inequality for a function whose derivative has a positive real part of order $α$

Deekonda Vamshee Krishna, Thoutreddy Ramreddy (2015)

Mathematica Bohemica

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The objective of this paper is to obtain sharp upper bound for the function $f$ for the second Hankel determinant $| a_{2} a_{4} - a_{3}^{2} |$ , when it belongs to the class of functions whose derivative has a positive real part of order $α$ $(0 \leq α < 1)$ , denoted by $R T (α)$ . Further, an upper bound for the inverse function of $f$ for the nonlinear functional (also called the second Hankel functional), denoted by $| t_{2} t_{4} - t_{3}^{2} |$ , was determined when it belongs to the same class of functions, using Toeplitz determinants.

A note on the functions $(S_{n} f) (z)$ defined by Ruscheweyn

S. Owa, C. Y. Shen (1988)

Matematički Vesnik

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On the powers of quasihomogeneous Toeplitz operators

Aissa Bouhali, Zohra Bendaoud, Issam Louhichi (2021)

Czechoslovak Mathematical Journal

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We present sufficient conditions for the existence of $p$ th powers of a quasihomogeneous Toeplitz operator $T_{e^{i s θ} ψ}$ , where $ψ$ is a radial polynomial function and $p$ , $s$ are natural numbers. A large class of examples is provided to illustrate our results. To our best knowledge those examples are not covered by the current literature. The main tools in the proof of our results are the Mellin transform and some classical theorems of complex analysis.

On products of some Toeplitz operators on polyanalytic Fock spaces

Irène Casseli (2020)

Czechoslovak Mathematical Journal

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The purpose of this paper is to study the Sarason’s problem on Fock spaces of polyanalytic functions. Namely, given two polyanalytic symbols $f$ and $g$ , we establish a necessary and sufficient condition for the boundedness of some Toeplitz products $T_{f} T_{\bar{g}}$ subjected to certain restriction on $f$ and $g$ . We also characterize this property in terms of the Berezin transform.

Strong maximum and minimum principles for parabolic functional-differential problems with nonlocal inequalities $[u^{j} (t_{0}, x) - K^{j}] + Σ_{i} h_{i} (x) [u^{j} (T_{i}, x) - K^{j}] \leq (\geq) 0$

Ludwik Byszewski (1990)

Annales Polonici Mathematici

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The initial value problem for parabolic equations with data in $L^{p} (R^{n})$

Eugene Fabes (1972)

Studia Mathematica

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On certain subclasses of analytic functions associated with the Carlson–Shaffer operator

Jagannath Patel, Ashok Kumar Sahoo (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The object of the present paper is to solve Fekete-Szego problem and determine the sharp upper bound to the second Hankel determinant for a certain class $R^{λ} (a, c, A, B)$ of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass ${\tilde{R}}^{λ} (a, c, A, B)$ of $R^{λ} (a, c, A, B)$ and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.

Existence results for a class of nonlinear parabolic equations with two lower order terms

Ahmed Aberqi, Jaouad Bennouna, M. Hammoumi, Mounir Mekkour, Ahmed Youssfi (2014)

Applicationes Mathematicae

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We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧ $\partial (e^{β u} - 1) / \partial t - {d i v (| \nabla u |}^{p - 2} \nabla u) + d i v (c (x, t) {| u |}^{s - 1} u) + b (x, t) {| \nabla u |}^{r} = f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{β u} - 1) (x, 0) = (e^{β u ₀} - 1) (x)$ in Ω. with s = (N+2)/(N+p) (p-1), $c (x, t) \in {(L^{τ} (Q T))}^{N}$ , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b (x, t) \in L^{N + 2, 1} (Q T)$ and f ∈ L¹(Q).

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

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Consider the nonlinear heat equation (E): $u_{t} - Δ u = {| u |}^{p - 1} u + b {| \nabla u |}^{q}$ . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C ₁ {(T - t)}^{- 1 / (p - 1)} {\leq | | u (t) | |}_{\infty} \leq C ₂ {(T - t)}^{- 1 / (p - 1)}$ . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality $u_{t} - u_{x x} \geq u^{p}$ . More general inequalities of the form $u_{t} - u_{x x} \geq f (u)$ with, for instance, $f (u) = (1 + u) l o g^{p} (1 + u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions...

On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations

P. Besala (1971)

Annales Polonici Mathematici

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Compression of slant Toeplitz operators on the Hardy space of $n$ -dimensional torus

Gopal Datt, Shesh Kumar Pandey (2020)

Czechoslovak Mathematical Journal

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This paper studies the compression of a $k$ th-order slant Toeplitz operator on the Hardy space $H^{2} (𝕋^{n})$ for integers $k \geq 2$ and $n \geq 1$ . It also provides a characterization of the compression of a $k$ th-order slant Toeplitz operator on $H^{2} (𝕋^{n})$ . Finally, the paper highlights certain properties, namely isometry, eigenvalues, eigenvectors, spectrum and spectral radius of the compression of $k$ th-order slant Toeplitz operator on the Hardy space $H^{2} (𝕋^{n})$ of $n$ -dimensional torus $𝕋^{n}$ .

Region of variability for functions with positive real part

Saminathan Ponnusamy, Allu Vasudevarao (2010)

Annales Polonici Mathematici

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For γ ∈ ℂ such that |γ| < π/2 and 0 ≤ β < 1, let $_{γ, β}$ denote the class of all analytic functions P in the unit disk with P(0) = 1 and $R e (e^{i γ} P (z)) > β c o s γ$ in . For any fixed z₀ ∈ and λ ∈ ̅, we shall determine the region of variability $V_{} (z ₀, λ)$ for $\int_{0}^{z ₀} P (ζ) d ζ$ when P ranges over the class $(λ) = P \in_{γ, β} : P^{'} (0) = 2 (1 - β) λ e^{- i γ} c o s γ .$ As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.

Generalized problem of starlikeness for products of close-to-star functions

Jacek Dziok (2013)

Annales Polonici Mathematici

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We consider functions of the type $F (z) = z \prod_{j = 1}^{n} {[f_{j} (z) / z]}^{a_{j}}$ , where $a_{j}$ are real numbers and $f_{j}$ are $β_{j}$ -strongly close-to-starlike functions of order $α_{j}$ . We look for conditions on the center and radius of the disk (a,r) = z:|z-a| < r, |a| < r ≤ 1 - |a|, ensuring that F((a,r)) is a domain starlike with respect to the origin.

Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda (2002)

Colloquium Mathematicae

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We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_{t} \geq - {(- Δ)}^{β / 2} u - V (x) u + h (x, t) u^{p}$ , where ${(- Δ)}^{β / 2}$ , 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying ${V ₊ (x) \sim a | x |}^{- b}$ , where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0....

Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space

Xing-Tang Dong, Ze-Hua Zhou (2013)

Studia Mathematica

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We present here a quite unexpected result: If the product of two quasihomogeneous Toeplitz operators $T_{f} T_{g}$ on the harmonic Bergman space is equal to a Toeplitz operator $T_{h}$ , then the product $T_{g} T_{f}$ is also the Toeplitz operator $T_{h}$ , and hence $T_{f}$ commutes with $T_{g}$ . From this we give necessary and sufficient conditions for the product of two Toeplitz operators, one quasihomogeneous and the other monomial, to be a Toeplitz operator.

Some properties for $α$ -starlike functions with respect to $k$ -symmetric points of complex order

H. E. Darwish, A. Y. Lashin, S. M. Sowileh (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In the present work, we introduce the subclass $𝒯_{γ, α}^{k} (ϕ)$ , of starlike functions with respect to $k$ -symmetric points of complex order $γ$ ( $γ \neq 0$ ) in the open unit disc . Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained.

Schatten class generalized Toeplitz operators on the Bergman space

Chunxu Xu, Tao Yu (2021)

Czechoslovak Mathematical Journal

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Let $μ$ be a finite positive measure on the unit disk and let $j \geq 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{μ}^{(j)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{μ}^{(j)}$ to be in the Schatten $p$ -class for $1 \leq p < \infty$ on the Bergman space $A^{2}$ , and then give a sufficient condition for $T_{μ}^{(j)}$ to be in the Schatten $p$ -class $(0 < p < 1)$ on $A^{2}$ . We also discuss the generalized Toeplitz operators with general bounded symbols. If $ϕ \in L^{\infty} (D, d A)$ and $1 < p < \infty$ , we define the generalized...