Asymptotic estimates for analytic functions in strips and their derivatives

  • G. I. Beregova Lviv Politechnic National University Lviv, Ukraine
  • S. I. Fedynyak Ukrainian Catholic University Lviv, Ukraine
  • Petro Filevych Lviv Polytechnic National University

Анотація

Let $-\infty\le A_0< A\le +\infty$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have $x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma):\sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, ${\Phi}_*(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$, and $F$ be an analytic function in the strip $\{s\in\mathbb{C}\colon A_0<\operatorname{Re}s<A\}$ such that the quantity $S(\sigma,F)=\sup\{|F(\sigma+it)|\colon t\in\mathbb{R}\}$ is finite for all $\sigma\in(A_0,A)$ and $F(s)\not\equiv0$. It is proved that if

\smallskip\centerline{$\ln S(\sigma,F)\le(1+o(1)\Phi(\sigma)$ as $\sigma\uparrow A$,}

\smallskip\noi then

\centerline{$\displaystyle
\varlimsup_{\sigma\uparrow A}\frac{S(\sigma,F')}{S(\sigma,F){\Phi}_*^{-1}(\sigma)}\le c_0,
$}

\smallskip\noi
where $c_0<1,1276$ is an absolute constant. From previously obtained results it follows that $c_0$ cannot be replaced by a constant less than $1$.

Біографії авторів

G. I. Beregova, Lviv Politechnic National University Lviv, Ukraine

Lviv Politechnic National University
Lviv, Ukraine

S. I. Fedynyak, Ukrainian Catholic University Lviv, Ukraine

Ukrainian Catholic University
Lviv, Ukraine

Petro Filevych, Lviv Polytechnic National University

Lviv Polytechnic National University

Посилання

S. Bernstein, Le¸cons sur les Propri´et´es Extr´emales et la Meilleure Approximation des Fonctions Analytiques

d’une Variable R´eelle, Paris: Gauthier-Villars, 1926.

T. Kovari, A note on entire functions, Acta Math. Acad. Sci. Hung. 8 (1957), №1–2, 87–90. doi: 10.1007/BF02025233

M.N. Sheremeta, Derivative of an entire function, Ukr. Math. J., 40 (1988), 188–192. doi: 10.1007/BF01056474

M.N. Sheremeta, S.I. Fedynyak, On the derivative of a Dirichlet series, Sib. Math. J., 39 (1998), №1, 181–197. doi: 10.1007/BF02732373

S.I. Fedynyak, P.V. Filevych, Growth estimates for a Dirichlet series and its derivative, Mat. Stud., 53 (2020), №1, 3–12. doi: 10.30970/ms.53.1.3-12

S.I. Fedynyak, P.V. Filevych, Growth estimates for the maximal term and central exponent of the derivative of a Dirichlet series, Carpathian Math. Publ., 12 (2020), №2, 269–279. doi: 10.15330/cmp.12.2.269-279

T.Ya. Hlova, P.V. Filevych, Generalized types of the growth of Dirichlet series, Carpathian Math. Publ., 7 (2015), №2, 172–187. doi: 10.15330/cmp.7.2.172-187

P.V. Filevych, Wiman–Valiron type inequalities for entire and random entire functions of finite logarithmic order, Sib. Math. J., 43 (2001), №3, 579–586. doi: 10.1023/A:1010435512666

P.V. Filevych, On influence of the arguments of coefficients of a power series expansion of an entire function on the growth of the maximum of its modulus, Sib. Math. J., 44 (2003), №3, 529–538. doi: 10.1023/A:1023825117420

T.Ya. Hlova, P.V. Filevych, The growth of analytic functions in the terms of generalized types, J. Lviv Politech. Nat. Univ., Physical and mathematical sciences, (2014), №804, 75–83. (in Ukrainian)

O.V. Shapovalovs’kyi, A remark on the derivative of entire Dirichlet series, Mat. Stud., 19 (2003), №1, 42–44.

G. Doetsch, ¨ Uber die obere Grenze des absoluten Betrages einer analytischen Funktion auf Geraden, Math. Z., 8 (1920), №3-4, 237–240. doi: 10.1007/BF01206529

S.I. Fedynyak, P.V. Filevych, Distance between a maximum modulus point and zero set of an analytic function, Mat. Stud., 52 (2019), №1, 10–23. doi: 10.30970/ms.52.1.10-23

Опубліковано
2022-06-27
Як цитувати
Beregova, G. I., Fedynyak, S. I., & Filevych, P. (2022). Asymptotic estimates for analytic functions in strips and their derivatives. Математичні студії, 57(2), 137-146. https://doi.org/10.30970/ms.57.2.137-146
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