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

S. Owa; C. Y. Shen

Displaying similar documents to “A note on the functions $(S_{n} f) (z)$ defined by Ruscheweyn”

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.

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.

Convolution conditions for bounded $α$ -starlike functions of complex order

A. Y. Lashin (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $A$ be the class of analytic functions in the unit disc $U$ of the complex plane $ℂ$ with the normalization $f (0) = f^{^{'}} (0) - 1 = 0$ . We introduce a subclass $S_{M}^{*} (α, b)$ of $A$ , which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class $S_{M}^{*} (n, α, b)$ ( $n \geq 0$ ) related to $S_{M}^{*} (α, b)$ is also considered under the same conditions. Among other things, we find convolution conditions for a function $f \in A$ to belong to the class $S_{M}^{*} (α, b)$ . Several properties of the class $S_{M}^{*} (n, α, b)$ are investigated. ...

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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We consider the class $ℋ_{0}$ of sense-preserving harmonic functions $f = h + \bar{g}$ defined in the unit disk $| z | < 1$ and normalized so that $h (0) = 0 = h^{'} (0) - 1$ and $g (0) = 0 = g^{'} (0)$ , where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $𝒫_{H}^{0} (α)$ and $𝒢_{H}^{0} (β)$ of functions from $ℋ_{0}$ and show that if $f \in 𝒫_{H}^{0} (α)$ and $F \in 𝒢_{H}^{0} (β)$ , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $α$ and $β$ are satisfied. In the second part we study the harmonic sections...

An integral operator on the classes $𝒮^{*} (α)$ and $𝒞𝒱ℋ (β)$

Nicoleta Ularu, Nicoleta Breaz (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The purpose of this paper is to study some properties related to convexity order and coefficients estimation for a general integral operator. We find the convexity order for this operator, using the analytic functions from the class of starlike functions of order $α$ and from the class $𝒞𝒱ℋ (β)$ and also we estimate the first two coefficients for functions obtained by this operator applied on the class $𝒞𝒱ℋ (β)$ .

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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Properties of functions concerned with Caratheodory functions

Mamoru Nunokawa, Emel Yavuz Duman, Shigeyoshi Owa (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $𝒫_{n}$ denote the class of analytic functions $p (z)$ of the form $p (z) = 1 + c_{n} z^{n} + c_{n + 1} z^{n + 1} + \dots$ in the open unit disc $𝕌$ . Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for $p (z)$ concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

On certain general integral operators of analytic functions

B. A. Frasin (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain new sufficient conditions for the operators $F_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ and $G_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ to be univalent in the open unit disc $𝒰$ , where the functions $f_{1}, f_{2}, . . ., f_{n}$ belong to the classes $S^{*} (a, b)$ and $𝒦 (a, b)$ . The order of convexity for the operators $F_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ and $G_{α_{1}, α_{2}, . . ., α_{n}, β} (z)$ is also determined. Furthermore, and for $β = 1$ , we obtain sufficient conditions for the operators $F_{n} (z)$ and $G_{n} (z)$ to be in the class $𝒦 (a, b)$ . Several corollaries and consequences of the main results are also considered.

On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space

Daniel Azagra, Mar Jiménez-Sevilla (2002)

Bulletin de la Société Mathématique de France

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We study the size of the sets of gradients of bump functions on the Hilbert space $ℓ_{2}$ , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in $ℓ_{2}$ can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space $ℓ_{2}$ can be uniformly approximated by $C^{1}$ smooth Lipschitz functions $ψ$ so that the cones generated by the ranges of its derivatives $ψ^{'} (ℓ_{2})$ have empty interior. This implies that there are $C^{1}$ smooth...

On spirallike functions of order $α$ and type $β$ with fixed second coefficient

M. K. Aouf (1988)

Matematički Vesnik

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A characterization of the disc $D^{n}$

Th. Friedrich (1974)

Colloquium Mathematicae

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Product property for capacities in $ℂ^{N}$

Mirosław Baran, Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

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The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{ν} (E ₁ \times E ₂) = m i n (C_{ν ₁} (E ₁), C_{ν ₂} (E ₂))$ , where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N ₁ + N ₂}$ , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$ , denote by C(E) the standard L-capacity and by $ω_{E}$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

A characterization of sets in $ℝ^{2}$ with DC distance function

Dušan Pokorný, Luděk Zajíček (2022)

Czechoslovak Mathematical Journal

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We give a complete characterization of closed sets $F \subset ℝ^{2}$ whose distance function $d_{F} : = dist (\cdot, F)$ is DC (i.e., is the difference of two convex functions on $ℝ^{2}$ ). Using this characterization, a number of properties of such sets is proved.

The general methods of finding the sum $S_{n}$ for all kinds of series, II

K. Orlov (1981)

Matematički Vesnik

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Operations between sets in geometry

Richard J. Gardner, Daniel Hug, Wolfgang Weil (2013)

Journal of the European Mathematical Society

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An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$ -dimensional Euclidean space $ℝ^{n}$ . It is proved that if $n \geq 2$ , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $G L (n)$ covariant, and associative if and only if it is $L_{p}$ addition for some $1 \leq p \leq \infty$ . It is also demonstrated...