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Analysis Mathematica
Author:
А. Ф. Леонтьев
LetD be an infinite convex domain and letF(z) bean analytic function inD. It is proved that there exist a functionf(z) regular inD and continuous in¯D+ (except for infinity) and an entire function
of growth order at most 1 and of minimal type, such that
\documentclass{aastex}
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\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$F(z) = \sum\limits_{k = 0}^\infty {c_k f^{(k)} (z), z \in D.}$$
\end{document}
of growth order at most 1 and of minimal type, such that
Analysis Mathematica
Author:
А. Ф. ЛЕОНТЬЕВ
The class
is introduced and thoroughly studied in the paper. By definition,H∈
if there exist sequences {А
n
} and {μ
n
}, ¦μ
n
¦ ↑ ∞ (depending onH(ϕ)) such that
\documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$\mathop {\lim \sup }\limits_{t \to \infty } \frac{{\ln \Phi \left( {re^{i\varphi } } \right)}}{{r^{\varrho _1 } }} = H\left( \varphi \right), \Phi \left( z \right) = \mathop \Sigma \limits_{k = 1}^\infty \left| {A_k E_\varrho \left( {\lambda _k z} \right)} \right|,$$
\end{document}
whereE
ϱ
(z) is a Mittag—Leffler function andϱ
1>ϱ>1/2. The significance of the class
is confirmed by the following theorem. For each functionH∈
there exists a sequence {λ
n
} with the following property: every entire functionF(z) of orderϱ
1 with the growth indicatorh
F
(ϕ)< <H(ϕ) can be expanded into the series
\documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$F\left( z \right) = \mathop \Sigma \limits_{n = 1}^\infty a_n E_\varrho \left( {\lambda _n z} \right),$$
\end{document}
furthermore,
\documentclass{aastex}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{mathrsfs}
\usepackage{pifont}
\usepackage{stmaryrd}
\usepackage{textcomp}
\usepackage{upgreek}
\usepackage{portland,xspace}
\usepackage{amsmath,amsxtra}
\pagestyle{empty}
\DeclareMathSizes{10}{9}{7}{6}
\begin{document}
$$\mathop {\lim sup}\limits_{r \to \infty } \frac{{\ln \Phi \left( {re^{i\varphi } } \right)}}{{r^{\varrho 1} }}< H\left( \varphi \right), \Phi \left( z \right) = \mathop \Sigma \limits_{n = 1}^\infty \left| {a_n E_\varrho \left( {\lambda _n z} \right)} \right|.$$
\end{document}
is introduced and thoroughly studied in the paper. By definition,H∈
if there exist sequences {А
n
} and {μ
n
}, ¦μ
n
¦ ↑ ∞ (depending onH(ϕ)) such that
is confirmed by the following theorem. For each functionH∈
there exists a sequence {λ
n
} with the following property: every entire functionF(z) of orderϱ
1 with the growth indicatorh
F
(ϕ)< <H(ϕ) can be expanded into the series
