Third Hankel determinant for starlike and convex functions with respect to symmetric points

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

Displaying similar documents to “Third Hankel determinant for starlike and convex functions with respect to symmetric points”

Coefficient inequality for transforms of parabolic starlike and uniformly convex functions

D. Vamshee Krishna, B. Venkateswarlu, T. RamReddy (2014)

Annales mathématiques Blaise Pascal

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The objective of this paper is to obtain sharp upper bound to the second Hankel functional associated with the $k^{t h}$ root transform ${[f (z^{k})]}^{\frac{1}{k}}$ of normalized analytic function $f (z)$ belonging to parabolic starlike and uniformly convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.

Slant Hankel operators

Subhash Chander Arora, Ruchika Batra, M. P. Singh (2006)

Archivum Mathematicum

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In this paper the notion of slant Hankel operator $K_{ϕ}$ , with symbol $ϕ$ in $L^{\infty}$ , on the space $L^{2} (𝕋)$ , $𝕋$ being the unit circle, is introduced. The matrix of the slant Hankel operator with respect to the usual basis ${z^{i} : i \in ℤ}$ of the space $L^{2}$ is given by $〈 α_{i j} 〉 = 〈 a_{- 2 i - j} 〉$ , where $\sum_{i = - \infty}^{\infty} a_{i} z^{i}$ is the Fourier expansion of $ϕ$ . Some algebraic properties such as the norm, compactness of the operator $K_{ϕ}$ are discussed. Along with the algebraic properties some spectral properties of such operators are discussed. Precisely, it is proved that for...

Bounded Toeplitz and Hankel products on weighted Bergman spaces of the unit ball

Małgorzata Michalska, Maria Nowak, Paweł Sobolewski (2010)

Annales Polonici Mathematici

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We prove a sufficient condition for products of Toeplitz operators $T_{f} T_{ḡ}$ , where f,g are square integrable holomorphic functions in the unit ball in ℂⁿ, to be bounded on the weighted Bergman space. This condition slightly improves the result obtained by K. Stroethoff and D. Zheng. The analogous condition for boundedness of products of Hankel operators $H_{f} H *_{g}$ is also given.

Sufficient conditions for starlike and convex functions

S. Ponnusamy, P. Vasundhra (2007)

Annales Polonici Mathematici

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For n ≥ 1, let denote the class of all analytic functions f in the unit disk Δ of the form $f (z) = z + \sum_{k = 2}^{\infty} a_{k} z^{k}$ . For Re α < 2 and γ > 0 given, let (γ,α) denote the class of all functions f ∈ satisfying the condition |f’(z) - α f(z)/z + α - 1| ≤ γ, z ∈ Δ. We find sufficient conditions for functions in (γ,α) to be starlike of order β. A generalization of this result along with some convolution results is also obtained.

Hankel determinant for a class of analytic functions of complex order defined by convolution

S. M. El-Deeb, M. K. Aouf (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain the Fekete-Szego inequalities for the functions of complex order defined by convolution. Also, we find upper bounds for the second Hankel determinant $| a_{2} a_{4} - a_{3}^{2} |$ for functions belonging to the class $S_{γ}^{b} (g (z); A, B)$ .

On the Hankel determinants of close-to-convex univalent functions.

Noor, K.Inayat (1980)

International Journal of Mathematics and Mathematical Sciences

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Failure of Nehari's theorem for multiplicative Hankel forms in Schatten classes

Ole Fredrik Brevig, Karl-Mikael Perfekt (2015)

Studia Mathematica

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Ortega-Cerdà-Seip demonstrated that there are bounded multiplicative Hankel forms which do not arise from bounded symbols. On the other hand, when such a form is in the Hilbert-Schmidt class ₂, Helson showed that it has a bounded symbol. The present work investigates forms belonging to the Schatten classes between these two cases. It is shown that for every $p > {(1 - l o g π / l o g 4)}^{- 1}$ there exist multiplicative Hankel forms in the Schatten class $_{p}$ which lack bounded symbols. The lower bound on p is in a certain...

Carleson measures and Toeplitz operators on small Bergman spaces on the ball

Van An Le (2021)

Czechoslovak Mathematical Journal

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We study Carleson measures and Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip’s results from the unit disk of $ℂ$ to the unit ball of $ℂ^{n}$ . We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1 < p < \infty$ .

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.

Hankel forms and sums of random variables

Henry Helson (2006)

Studia Mathematica

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A well known theorem of Nehari asserts on the circle group that bilinear forms in H² can be lifted to linear functionals on H¹. We show that this result can be extended to Hankel forms in infinitely many variables of a certain type. As a corollary we find a new proof that all the $L^{p}$ norms on the class of Steinhaus series are equivalent.

Compact operators on the weighted Bergman space A¹(ψ)

Tao Yu (2006)

Studia Mathematica

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We show that a bounded linear operator S on the weighted Bergman space A¹(ψ) is compact and the predual space A₀(φ) of A¹(ψ) is invariant under S* if and only if $S k_{z} \to 0$ as z → ∂D, where $k_{z}$ is the normalized reproducing kernel of A¹(ψ). As an application, we give conditions for an operator in the Toeplitz algebra to be compact.

On Pták’s generalization of Hankel operators

Carmen H. Mancera, Pedro José Paúl (2001)

Czechoslovak Mathematical Journal

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In 1997 Pták defined generalized Hankel operators as follows: Given two contractions $T_{1} \in ℬ (ℋ_{1})$ and $T_{2} \in ℬ (ℋ_{2})$ , an operator $X ℋ_{1} \to ℋ_{2}$ is said to be a generalized Hankel operator if $T_{2} X = X T_{1}^{*}$ and $X$ satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of $T_{1}$ and $T_{2}$ . This approach, call it (P), contrasts with a previous one developed by Pták and Vrbová in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong...

Uniformly starlike functions and uniformly convex functions related to the Pascal distribution

Gangadharan Murugusundaramoorthy, Sibel Yalçın (2021)

Mathematica Bohemica

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In this article, we aim to find sufficient conditions for a convolution of analytic univalent functions and the Pascal distribution series to belong to the families of uniformly starlike functions and uniformly convex functions in the open unit disk $𝕌$ . We also state corollaries of our main results.

Reducing subspaces of Toeplitz operators on Dirichlet type spaces of the bidisk

Hongzhao Lin, Zhongming Teng (2021)

Czechoslovak Mathematical Journal

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The reducing subspaces of Toeplitz operators $T_{z_{1}^{N} {\bar{z}}_{2}^{M}}$ on Dirichlet type spaces of the $𝒟_{α} (𝔻^{2})$ are described, which extends the results for the corresponding operators on Bergman spaces of the bidisk.