TY - JOUR
AU - Skaskiv, O.B.
AU - Kuryliak, A.O.
PY - 2020/12/30
Y2 - 2024/07/23
TI - Wiman's type inequality for analytic and entire functions and $h$-measure of an exceptional sets
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 12
IS - 2
SE - Scientific articles
DO - 10.15330/cmp.12.2.492-498
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/4393
SP - 492-498
AB - <p>Let $\mathcal{E}_R$ be the class of analytic functions $f$ represented by power series of the form $f(z)=\sum\limits\limits_{n=0}^{+\infty}a_n z^n$ with the radius of convergence $R:=R(f)\in(0;+\infty].$ For $r\in [0, R)$ we denote the maximum modulus by $M_f(r)=\max\{|f(z)|\colon$ $ |z|=r\}$ and the maximal term of the series by $\mu_f(r)=\max\{|a_n| r^n\colon n\geq 0\}$. We also denote by $\mathcal{H}_R$, $R\leq +\infty$, the class of continuous positive functions, which increase on $[0;R)$ to $+\infty$, such that $h(r)\geq2$ for all $r\in (0,R)$ and $ \int^R_{r_{0}} h(r) d\ln r =+\infty $ for some $r_0\in(0,R)$. In particular, the following statements are proved.</p><p>$1^0.$ If $h\in \mathcal{H}_R$ and $f\in \mathcal{E}_R,$ then for any $\delta>0$ there exist $E(\delta,f,h):=E\subset(0,R)$, $r_0 \in (0,R)$ such that $$ \forall\ r\in (r_0,R)\backslash E\colon\ M_f(r)\leq h(r) \mu_f(r) \big\{\ln h(r)\ln(h(r)\mu_f(r))\big\}^{1/2+\delta}$$ and $$\int
olimits_E h(r) d\ln r < +\infty. $$</p><p>$2^0.$ If we additionally assume that the function $f\in \mathcal{E}_R$ is unbounded, then $$ \ln M_f(r)\leq(1+o(1))\ln (h(r)\mu_f(r)) $$ holds as $r\to R$, $r
otin E$.</p><p>Remark, that assertion $1^0$ at $h(r)\equiv \text{const}$ implies the classical Wiman-Valiron theorem for entire functions and at $h(r)\equiv 1/(1-r)$ theorem about the Kövari-type inequality for analytic functions in the unit disc. From statement $2^0$ in the case that $\ln h(r)=o(\ln\mu_f(r))$, $r\to R$, it follows that $ \ln M_f(r)=(1+o(1))\ln \mu_f(r) $ holds as $r\to R$, $r
otin E$.</p>
ER -