Local analytic solutions of the functional equation $Φ(z) = H(z; Φ[f_1(z)],...,Φ[f_m(z)])$

Wilhelmina Smajdor

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Displaying similar documents to “Local analytic solutions of the functional equation $Φ (z) = H (z; Φ [f_{1} (z)], . . ., Φ [f_{m} (z)])$ ”

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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.

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Annales Polonici Mathematici

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This work deals with Feigenbaum’s functional equation ⎧ $h (g (x)) = g^{p} (h (x))$ , ⎨ ⎩ g(0) = 1, -1 ≤ g(x) ≤ 1, x∈[-1,1] where p ≥ 2 is an integer, $g^{p}$ is the p-fold iteration of g, and h is a strictly monotone odd continuous function on [-1,1] with h(0) = 0 and |h(x)| < |x| (x ∈ [-1,1], x ≠ 0). Using a constructive method, we discuss the existence of continuous unimodal even solutions of the above equation.

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.

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We prove that any divisor $Y$ of a global analytic set $X \subset ℝ^{n}$ has a generic equation, that is, there is an analytic function vanishing on $Y$ with multiplicity one along each irreducible component of $Y$ . We also prove that there are functions with arbitrary multiplicities along $Y$ . The main result states that if $X$ is pure dimensional, $Y$ is locally principal, $X / Y$ is not connected and $Y$ represents the zero class in $H_{q - 1}^{\infty} (X, ℤ_{2})$ then the divisor $Y$ is globally principal.

On a problem concerning quasianalytic local rings

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Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f \in_{m}$ and f̂ be its Taylor series at $0 \in ℝ^{m}$ . Split the set $ℕ^{m}$ of exponents into two disjoint subsets A and B, $ℕ^{m} = A \cup B$ , and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist $g, h \in_{m}$ with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some...

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Plane $d$ -webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two $d$ -webs, $d \geq 3$ are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable $d$ -webs, however, there always exists a nonmeasurable conjugacy;...

La Queste del Saint $G R_{a}$ (AL): a Computational Approach to Local Algebra

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

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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 Pólya's Theorem in several complex variables

Ozan Günyüz, Vyacheslav Zakharyuta (2015)

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Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f (z) = \sum_{k = 0}^{\infty} a_{k} z^{- k - 1}$ be its Taylor expansion at ∞, and $H_{s} (f) = d e t {(a_{k + l})}_{k, l = 0}^{s}$ the sequence of Hankel determinants. The classical Pólya inequality says that $l i m s u p_{s \to \infty} {| H_{s} (f) |}^{1 / s ²} \leq d (K)$ , where d(K) is the transfinite diameter of K. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Pólya’s inequality, considered by the second author in Math. USSR Sbornik 25 (1975), 350-364.

The functional equation $f^{2} (x) = g (x)$

James C. Lillo (1967)

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On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ , = φ: ℝ → ℝ|φ is continuous, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the...

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A. Makowski (1964)

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Let $Ω \subset ℂ^{n}$ be a bounded, simply connected $ℂ$ -convex domain. Let $α \in ℤ_{+}^{n}$ and let $f$ be a function on $Ω$ which is separately $C^{2 α_{j} - 1}$ -smooth with respect to $z_{j}$ (by which we mean jointly $C^{2 α_{j} - 1}$ -smooth with respect to $Re z_{j}$ , $Im z_{j}$ ). If $f$ is $α$ -analytic on $Ω ∖ f^{- 1} (0)$ , then $f$ is $α$ -analytic on $Ω$ . The result is well-known for the case $α_{i} = 1$ , $1 \leq i \leq n$ , even when $f$ a priori is only known to be continuous.

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

Th. Friedrich (1974)

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Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

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Displaying similar documents to “Local analytic solutions of the functional equation Φ ( z ) = H ( z ; Φ [ f 1 ( z ) ] , . . . , Φ [ f m ( z ) ] ) ”

Displaying similar documents to “Local analytic solutions of the functional equation $Φ (z) = H (z; Φ [f_{1} (z)], . . ., Φ [f_{m} (z)])$ ”