부산대학교 도서관

Let $\Bbb{D}=\{z\in\Bbb{C}:|z|<1\}$ denote the unit disk in the complex plane $\Bbb{C}$. Also, let $\Cal{B}$ be the class of all analytic functions $f$ defined on $\Bbb{D}$ such that $|f(z)|\leq1$, $z\in\Bbb{D}$. In 1914, it was discovered by H. Bohr [Proc. London Math. Soc. (2) {\bf 13} (1914), 1--5; MR1577494] that if $f(z)=\sum_{n=0}^{\infty} a_nz^n\in\Bbb{B}$, then $$ \sum_{n=0}^{\infty} |a_nz^n|\leq 1 $$ for $|z|\leq 1/3$ and the constant $1/3$ cannot be improved. The inequality is known as the Bohr inequality while the constant $1/3$ is known as the Bohr radius of the class $\Cal{B}$. Recently, the concept of Bohr radius has been extended to the study of integral operators, which are defined on certain analytic function classes. In this work, the authors manage to obtain two Bohr-type radii by considering the $\beta$-Cesàro operator $T_{\beta}$ and the Bernadi operator $L_{\gamma}$. \par In 2020, the Cesàro operator $T$ on the class $\Cal{B}$, as defined in [G.~H. Hardy and J.~E. Littlewood, Math. Z. {\bf 34} (1932), no.~1, 403--439; MR1545260], $$ T[f](z)=\int_0^1\frac{f(tz)}{1-tz}\,dt=\sum_{n=0}^{\infty} \left(\frac{1}{n+1}\sum_{k=0}^n a_k\right)z^n, $$ was first applied to the study of Bohr radii in [I.~R. Kayumov, D.~M. Khammatova and S. Ponnusamy, C. R. Math. Acad. Sci. Paris {\bf 358} (2020), no.~5, 615--620; MR4149861]. Motivated by this work, the authors investigate the Bohr radius for a generalized version of Cesàro operator. They introduce the $\beta$-Cesàro operator $T_{\beta}$ ($\beta>0$), which is defined on the class $\Cal{B}$ and given by $$ T_{\beta}[f](z)=\int_0^1\frac{f(tz)}{(1-tz)^{\beta}}\,dt = \sum_{n=0}^{\infty} \left(\frac{1}{n+1}\sum_{k=0}^n \frac{\Gamma(n-k+\beta)}{\Gamma(n-k+1)\Gamma(\beta)} a_k\right)z^n,\quad z\in\Bbb{D}, $$ for $f(z)=\sum_{n=0}^{\infty} a_nz^n\in\Cal{B}$. It can be seen the $\beta$-Cesàro operator reduces to the classical Cesàro operator when $\beta=1$. The authors show that the following Bohr-type inequality $$ \sum_{n=0}^{\infty} \left(\frac{1}{n+1}\sum_{k=0}^n \frac{\Gamma(n-k+\beta)}{\Gamma(n-k+1)\Gamma(\beta)} |a_k|\right)r^n\leq \frac{1}{r}\left[\frac{1-(1-r)^{1-\beta}}{1-\beta}\right] \tag1 $$ holds for $r\leq R(\beta)$, where $R(\beta)$ is the positive root of the equation $$ \frac{3[1-(1-x)^{1-\beta}]}{1-\beta}-\frac{2[(1-x)^{-\beta}-1]}{\beta}=0. $$ The Möbius transformation $$ \phi_a(z)=\frac{z-a}{1-az}\in \Cal{B},\quad a\in[0,1), $$ can be used to verify that the constant $R(\beta)$ cannot be improved. The proof makes use of Wiener's estimate ($|a_n|\leq 1-a^2$, $n\geq 1$) to find a non-negative function $\phi(a)$ which dominates the infinite sum on the left of inequality (1). The authors then show that $\phi(a)$ is an increasing function in terms of $a$ and is dominated by the function on the right hand side of inequality (1). \par Let $\Cal{B}_0=\{f\in \Cal{B}:f(0)=0\}$ be the class of normalized functions in $\Cal{B}$. If $g(z)=\sum_{n=1}^{\infty} b_nz^n\in\Cal{B}_0$, then there is another form of $\beta$-Cesàro operator $C_{\beta}$ ($\beta>0$) defined on the class $\Cal{B}_0$ [S. Kumar and S.~K. Sahoo, Rocky Mountain J. Math. {\bf 50} (2020), no.~5, 1723--1746; MR4170682]: $$ C_{\beta}[g](z)=\int_0^1\frac{g(tz)}{t(1-tz)^{\beta}}\,dt= \sum_{n=0}^{\infty} \left(\frac{1}{n+1}\sum_{k=0}^n \frac{\Gamma(n-k+\beta)}{\Gamma(n-k+1)\Gamma(\beta)} b_{k+1}\right)z^{n+1},\quad z\in\Bbb{D}. $$ The authors remark that the Bohr radii for the operators $T_{\beta}$ and $C_{\beta}$ are exactly the same where the corresponding Bohr-type inequality is given by $$ \sum_{n=0}^{\infty} \left(\frac{1}{n+1}\sum_{k=0}^n \frac{\Gamma(n-k+\beta)}{\Gamma(n-k+1)\Gamma(\beta)} |b_{k+1}|\right)r^{n+1}\leq \frac{1}{r}\left[\frac{1-(1-r)^{1-\beta}}{1-\beta}\right]. $$ \par Last but not least, the authors also obtain Bohr-type radii for the Bernadi operator $L_{\gamma}$ [S.~S. Miller and P.~T. Mocanu, {\it Differential subordinations}, Monogr. Textbooks Pure Appl. Math., 225, Dekker, New York, 2000 (p. 11); MR1760285], which is defined on the class $\Cal{B}$ and given by $$ L_{\gamma}[f](z)\coloneq \sum_{n=m}^{\infty}\frac{a_n}{n+\gamma} z^n=\int_0^1f(zt) t^{\gamma-1}\,dt, $$ for $f(z)=\sum_{n=m}^{\infty} a_nz^n\in\Cal{B}$ and $\gamma>-m$. The authors prove that the following Bohr-type inequality $$ \sum_{n=m}^{\infty} \frac{|a_n|}{n+\gamma} r^n\leq \frac{1}{m+\gamma}r^m \tag2 $$ holds for $r\leq R(\gamma)$. The constant $R(\gamma)$ is a positive root of some equation and cannot be improved. In particular, if $\gamma=1$ and $m=0$, then the Bernadi operator $L_{\gamma}$ is reduced to the well-known Libera operator [see S.~S. Miller and P.~T. Mocanu, op. cit.; R. Parvatham, S. Ponnusamy and S.~K. Sahoo, Hiroshima Math. J. {\bf 38} (2008), no.~1, 19--29; MR2397377]. The techniques applied in the proof of this result are very much similar to the proof of Bohr-type radii for the Cesàro operator, and the Möbius transformation $\phi_a$ can be used to show that $R(\gamma)$ cannot be improved.