Bounded l-index and l-M-index and compositions of analytic
functions

A. I. Bandura; M. M. Sheremeta

doi:10.15330/ms.48.1.180-188

We partially proved a conjecture from Mat. Stud. 47 (2017), no.2, 207--210: for an entire function $f$ the function $H(z)\!=\!f({1}/{(1\!-\!z)^n})$, $n\in\mathbb{N}$, is of bounded $l$-index in $\mathbb{C}\setminus\{0\}$ with $l(|z|)={\beta}/{(1-|z|)^{n+1}}$, $\beta>1$, if and only if $f$ is of bounded index. Also the boundedness of $l$-$M$-index of the function $H$ is investigated. For arbitrary entire functions $f$ and $g$ the boundedness of the $l$-$M$-index of the function $F(z)=f(g(z))$ is studied with respect to boundedness of the $M$-index of a function $f$ with $l(r)=M'_g(r),$ $M_g(r)=\max\{|g(z)|\colon |z|=r\}.$

1. M. M. Sheremeta, On the l-index boundedness of some composition of functions, Mat. Stud., 47 (2017), №2, 207–210.

2. Sh. Abuarabi, M.M. Sheremeta, Entire functions of bounded l-M-index, Dopov. AN URSR, 11 (1989), 3–5. (in Ukrainian)

3. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), №1, 30–52. (in Ukrainian)

4. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded and unbounded index in direction, Mat. Stud., 27 (2007), №2, 211–215. (in Ukrainian)

5. A.I. Bandura, O.B. Skaskiv, Boundedness of L-index in direction of functions of the form f(.z;m.) and existence theorems, Mat. Stud., 41 (2014), №1, 45–52.

6. A.I. Bandura, O.B. Skaskiv, Directional logarithmic derivative and the distribution of zeros of an entire function of bounded L-Index along the direction, Ukrain. Mat. J., 69 (2017), №3, 500–508.

7. A. Bandura, O. Skaskiv, Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci., 227 (2017), №1.

8. A. Bandura, O. Skaskiv, Analytic functions in the unit ball. Bounded L-index in joint variables and solutions of systems of PDE’s, Beau-Bassin: LAP Lambert Academic Publishing, 2017, 100 p. https://arxiv.org/abs/1705.09568

9. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p.

10. A.I. Bandura, M.T. Bordulyak, O.B. Skaskiv, Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud., 45 (2016), №1, 12–26.

11. A.I. Bandura, N.V. Petrechko, O.B. Skaskiv, Analytic functions in a polydisc of bounded L-index in joint variables, Mat. Stud., 46 (2016), №1, 72–80.

12. A. Bandura A., N. Petrechko, O. Skaskiv. Maximum modulus in a bidisc of analytic functions of bounded L-index and an analogue of Hayman’s theorem, Matematica Bohemica, doi: 10.21136/MB.2017.0110-16 (in print)

13. A.I. Bandura, Composition of entire functions and bounded L-index in direction, Mat. Stud., 47 (2017), №2, 179–184.

14. J. Clunie, The composition of entire and meromorphic functions, In: Mathematical essays dedicated to A.J. Macintyre, Ohio Univ. Press, Athens, Ohio, 75–92 (1970).

15. G.H. Fricke, S.M. Shah, W.C. Sisarcick, A characterization of entire functions of exponential Type and M-bounded Index, Indiana Univ. Math. J., 23 (1974), №5, 405–412.

16. W.K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1973), №1, 117–137.

17. V.O. Kushnir, M.M. Sheremeta, Analytic functions of bounded l-index, Mat. Stud., 12, №1 (1999), 59–66.

18. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., Amer. Math. Soc.: Providence, Rhode Island, 2 (1968), 298–307.

19. Ya.V. Mykytyuk, S.I. Fedynyak, M.M. Sheremeta, Dirichlet series of bounded l .M-index, Mat. Stud., 11 (1999), №2, 159–166.

20. M.N. Sheremeta, Entire functions and Dirichlet series of bounded l-index, Russian Math. (Iz. VUZ), 36 (1992), №9, 76–82.

21. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p.

22. M.M. Sheremeta, On the growth of a composition of entire functions, Carpathian Math. Publ., 9 (2017), №2, 181–187.

	Bounded l-index and l-M-index and compositions of analytic functions
Author	A. I. Bandura, M. M. Sheremeta andriykopanytsia@gmail.com, m_m_sheremeta@gmail.com Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine; Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	We partially proved a conjecture from Mat. Stud. 47 (2017), no.2, 207--210: for an entire function $f$ the function $H(z)\!=\!f({1}/{(1\!-\!z)^n})$, $n\in\mathbb{N}$, is of bounded $l$-index in $\mathbb{C}\setminus\{0\}$ with $l(\|z\|)={\beta}/{(1-\|z\|)^{n+1}}$, $\beta>1$, if and only if $f$ is of bounded index. Also the boundedness of $l$-$M$-index of the function $H$ is investigated. For arbitrary entire functions $f$ and $g$ the boundedness of the $l$-$M$-index of the function $F(z)=f(g(z))$ is studied with respect to boundedness of the $M$-index of a function $f$ with $l(r)=M'_g(r),$ $M_g(r)=\max\{\|g(z)\|\colon \|z\|=r\}.$
Keywords	analytic function; entire function; bounded l-index; bounded index; bounded M-index; bounded l-M-index; composition of functions; growth estimate; punctured plane; disc; maximum modulus
DOI	doi:10.15330/ms.48.2.180-188
Reference	1. M. M. Sheremeta, On the l-index boundedness of some composition of functions, Mat. Stud., 47 (2017), №2, 207–210. 2. Sh. Abuarabi, M.M. Sheremeta, Entire functions of bounded l-M-index, Dopov. AN URSR, 11 (1989), 3–5. (in Ukrainian) 3. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), №1, 30–52. (in Ukrainian) 4. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded and unbounded index in direction, Mat. Stud., 27 (2007), №2, 211–215. (in Ukrainian) 5. A.I. Bandura, O.B. Skaskiv, Boundedness of L-index in direction of functions of the form f(.z;m.) and existence theorems, Mat. Stud., 41 (2014), №1, 45–52. 6. A.I. Bandura, O.B. Skaskiv, Directional logarithmic derivative and the distribution of zeros of an entire function of bounded L-Index along the direction, Ukrain. Mat. J., 69 (2017), №3, 500–508. 7. A. Bandura, O. Skaskiv, Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci., 227 (2017), №1. 8. A. Bandura, O. Skaskiv, Analytic functions in the unit ball. Bounded L-index in joint variables and solutions of systems of PDE’s, Beau-Bassin: LAP Lambert Academic Publishing, 2017, 100 p. https://arxiv.org/abs/1705.09568 9. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p. 10. A.I. Bandura, M.T. Bordulyak, O.B. Skaskiv, Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud., 45 (2016), №1, 12–26. 11. A.I. Bandura, N.V. Petrechko, O.B. Skaskiv, Analytic functions in a polydisc of bounded L-index in joint variables, Mat. Stud., 46 (2016), №1, 72–80. 12. A. Bandura A., N. Petrechko, O. Skaskiv. Maximum modulus in a bidisc of analytic functions of bounded L-index and an analogue of Hayman’s theorem, Matematica Bohemica, doi: 10.21136/MB.2017.0110-16 (in print) 13. A.I. Bandura, Composition of entire functions and bounded L-index in direction, Mat. Stud., 47 (2017), №2, 179–184. 14. J. Clunie, The composition of entire and meromorphic functions, In: Mathematical essays dedicated to A.J. Macintyre, Ohio Univ. Press, Athens, Ohio, 75–92 (1970). 15. G.H. Fricke, S.M. Shah, W.C. Sisarcick, A characterization of entire functions of exponential Type and M-bounded Index, Indiana Univ. Math. J., 23 (1974), №5, 405–412. 16. W.K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1973), №1, 117–137. 17. V.O. Kushnir, M.M. Sheremeta, Analytic functions of bounded l-index, Mat. Stud., 12, №1 (1999), 59–66. 18. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., Amer. Math. Soc.: Providence, Rhode Island, 2 (1968), 298–307. 19. Ya.V. Mykytyuk, S.I. Fedynyak, M.M. Sheremeta, Dirichlet series of bounded l .M-index, Mat. Stud., 11 (1999), №2, 159–166. 20. M.N. Sheremeta, Entire functions and Dirichlet series of bounded l-index, Russian Math. (Iz. VUZ), 36 (1992), №9, 76–82. 21. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p. 22. M.M. Sheremeta, On the growth of a composition of entire functions, Carpathian Math. Publ., 9 (2017), №2, 181–187.
Pages	180-188
Volume	48
Issue	2
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Bounded l-index and l-M-index and compositions of analytic functions