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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Ole Fredrik Brevig, Karl-Mikael Perfekt (2015)

Studia Mathematica

Similarity:

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...

Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space

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

Studia Mathematica

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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...

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Kent Griffin, Jeffrey L. Stuart, Michael J. Tsatsomeros (2008)

Czechoslovak Mathematical Journal

Similarity:

Let $a$ , $b$ and $c$ be fixed complex numbers. Let $M_{n} (a, b, c)$ be the $n \times n$ Toeplitz matrix all of whose entries above the diagonal are $a$ , all of whose entries below the diagonal are $b$ , and all of whose entries on the diagonal are $c$ . For $1 \leq k \leq n$ , each $k \times k$ principal minor of $M_{n} (a, b, c)$ has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of $M_{n} (a, b, c)$ . We also show that all complex polynomials in $M_{n} (a, b, c)$ are Toeplitz matrices. In particular, the inverse of $M_{n} (a, b, c)$ is a Toeplitz...

The essential spectrum of Toeplitz tuples with symbols in $H^{\infty} + C$

Jörg Eschmeier (2013)

Studia Mathematica

Similarity:

Let H²(D) be the Hardy space on a bounded strictly pseudoconvex domain D ⊂ ℂⁿ with smooth boundary. Using Gelfand theory and a spectral mapping theorem of Andersson and Sandberg (2003) for Toeplitz tuples with $H^{\infty}$ -symbol, we show that a Toeplitz tuple $T_{f} = (T_{f ₁}, . . ., T_{f ₘ}) \in L {(H ² (σ))}^{m}$ with symbols $f_{i} \in H^{\infty} + C$ is Fredholm if and only if the Poisson-Szegö extension of f is bounded away from zero near the boundary of D. Corresponding results are obtained for the case of Bergman spaces. Thus we extend results of McDonald (1977) and...