On zeros of derivatives of an entire function

M.T.Bordulyak; M.M.Sheremeta; Yu.S.Trukhan

For a finite system $S(f)=\{f^{(n_1)}, f^{(n_2)},\dots, f^{(n_k)}\}$ of derivatives of an entire\break transcedental function $f$ let $d_{S(f)}(z)$ be the radius of the largest disk with the center at $z$ in which any derivative of $S(f)$ does not vanish. Conditions under which $\sup\{d_{S(f)}(z):\,z\in {\Bbb C}\}=+\infty$ are investigated.

1. Sreenivasulu V. The depth of an entire function // Journal of the Indian Math. Soc. -- 1992. -- V. 58, No. 2. -- P. 105--116.

2. Frank G. Uber den Index einer ganzen Funktionen // Arch. der Math. -- 1971. -- Bd. 175--180.

3. Lepson B. Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index // Proc. Sympos. Pure Math. V.2. Amer. Math. Soc., Providence, Phode Island. - 1968. -- P. 298--307.

4. Shah S.M. Entire functions of bounded index // Lect. Notes in Math. -- 1977. -- V. 589. -- P. 117--145.

5. Korenkov M.…. On value distribution of sigma-function of Weierstrass // In the book: Matemat. sbornik. -- K.: Naukova dumka. -- 1976. -- P. 240--242.

6. Kuratowski K. Topology. V.II // M.: Mir. -- 1969. -- 624 p.

	On zeros of derivatives of an entire function
Author	M.T.Bordulyak, M.M.Sheremeta, Yu.S.Trukhan tftj@franko.lviv.ua Lviv Ivan Franko National University
Abstract	For a finite system $S(f)=\{f^{(n_1)}, f^{(n_2)},\dots, f^{(n_k)}\}$ of derivatives of an entire\break transcedental function $f$ let $d_{S(f)}(z)$ be the radius of the largest disk with the center at $z$ in which any derivative of $S(f)$ does not vanish. Conditions under which $\sup\{d_{S(f)}(z):\,z\in {\Bbb C}\}=+\infty$ are investigated.
Keywords	zero, entire function, derivative
DOI	doi:10.30970/ms.25.2.141-148
Reference	1. Sreenivasulu V. The depth of an entire function // Journal of the Indian Math. Soc. -- 1992. -- V. 58, No. 2. -- P. 105--116. 2. Frank G. Uber den Index einer ganzen Funktionen // Arch. der Math. -- 1971. -- Bd. 175--180. 3. Lepson B. Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index // Proc. Sympos. Pure Math. V.2. Amer. Math. Soc., Providence, Phode Island. - 1968. -- P. 298--307. 4. Shah S.M. Entire functions of bounded index // Lect. Notes in Math. -- 1977. -- V. 589. -- P. 117--145. 5. Korenkov M.…. On value distribution of sigma-function of Weierstrass // In the book: Matemat. sbornik. -- K.: Naukova dumka. -- 1976. -- P. 240--242. 6. Kuratowski K. Topology. V.II // M.: Mir. -- 1969. -- 624 p.
Pages	141-148
Volume	25
Issue	2
Year	2006
Journal	Matematychni Studii
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