On the l-index boundedness of some composition of functions

M. M. Sheremeta

doi:10.15330/ms.47.2.207-210

It is suggested that for an entire function $f$ the function $F(z)=f(\frac{q}{(1-z)^n})$, $n\in{\Bbb N}$, is of bounded $l$-index with $l(|z|)=\frac{\beta}{(1-|z|)^{n+1}}$, $\beta>1$, if and only if $f$ is of bounded index.

1. Sheremeta M.M. Analytic functions of bounded index. - Lviv: VNTL Publishes, 1999.

2. Goldberg A.A., Sheremeta M.N. Existence of an entire transcendental function of bounded l-index// Mat. Zametki. - 1995. - V.57, no.1 - P. 125-129 (in Russian); Engl. transl. Math Notes. - 1995. - V.57, no.1. - P. 88-90.

3. Sheremeta M.M., Bordulyak M.T. On the existence of entire functions of bounded l-index and l-regular growth// Ukrainian Math. Zh. - 1996. - V.48, no.9. - P. 1166-1182 (in Russian); Engl. transl.: Ukrainian Math. J. - 1996. - V.48, no.9. - P. 1322-1340.

4. Chyzhykov I.E., Sheremeta M.M. Boundedness of l-index for entire functions of zero genus// Mat. Stud. - 2001. - V.16, no.2. - P. 124-130.

5. Sheremeta M.M., Bordulyak M.T. Boundedness of the l-index of Laguerre-Polya entire functions// Ukrainian Math. Zh. - 2003. - V.55, no.1. - P. 91-99 (in Ukrainian); Engl. transl.: Ukrainian Math. J. - 2003. - V.55, no.1, 112-125.

6. Chyzhykov I.E., Sheremeta M.M. On the boundedness of l-index for entire functions of zero genus// Dopovidi NAN Ukraine. - 2003. - V.7. - P. 33-39. (in Ukrainian)

7. Goldberg A.A., Sheremeta M.M. On the boundedness of the l-index of canonical products// Ukr. Mat. Visn. - 2005. - V.2, no.1. - P. 52-64 (in Ukrainian); Eng. transl.: Ukr. Math. Bull. - 2005. - V.2, no.1. - P. 53-65.

8. Trukhan Yu.S., Sheremeta M.M. Boundedness of l-index of Blaschke product// Mat. Stud. - 2002. - V.17, no.2. - P. 127-137. (in Ukrainian)

9. Trukhan Yu.S., Sheremeta M.M. On l-index boundedness of the Blaschke product// Mat. Stud. - 2003. - V.19, no.1. - P. 106-112.

10. Sheremeta M.M., Trukhan Yu.S. Boundedness of the l-index of the Naftalevich-Tsuji product// Ukrainian Math. Zh. - 2004. - V.56, no.2. - P. 247-256 (in Ukrainian); Engl. transl.: Ukrainian Math. J. - 2004. - V.56, no.2. - P. 305-317.

11. Sheremeta M.N. Entire functions and Dirichlet series of bounded l-index// Izv. Vyssh. Uchebn. Zaved. Mat. - 1992. - V.9. - P. 81-87. (in Russian). Engl. transl.: Russian Math. (Iz. VUZ). - 1992. - V.36, no.9. - P. 76-82.

12. Strochyk S.M., Sheremeta M.M. Analytic in the unit disc functions of bounded index// Dopov. Akad. Nauk Ukr. - 1993. - no.1. - P. 19-22. (in Ukrainian)

13. Shah S.M. Univalence of a function f and its successive derivative when f satisfies a differential equation, II// J. Math. Anal. and Appl. - 1989. - V.142. - P. 422-430.

14. Hayman W.K. Differential inequalities and local valency// Pacific J. Math. - 1973. - V.44. - P. 117-137.

15. Sheremeta Z.M. Properties of derivatives of an entire solution of a differential equation// Mat. method. and fiz.-mech. polya. - 2006. - V.49, no.2. - P. 80-85. (in Ukrainian)

16. Sheremeta Z.M., Sheremeta M.M. Properties of entire solutions of differential equations// Ukrainian Math. Zh. - 2006. - V.58, no.12. - P. 1693-1703 (in Ukrainian); Engl. transl.: Ukrainian Math. J. - 2006. - V.56, no.12. - P. 1924-1934.

17. Sheremeta Z.M., Sheremeta M.M. On the l-index boundedness of entire solutions of a differential equation // Dopovidi NAN Ukraine. - 2007. - V.56, no.2. - P. 31-36. (in Ukrainian)

	On the l-index boundedness of some composition of functions
Author	M. M. Sheremeta m_m_sheremeta@gmail.com Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	It is suggested that for an entire function $f$ the function $F(z)=f(\frac{q}{(1-z)^n})$, $n\in{\Bbb N}$, is of bounded $l$-index with $l(\|z\|)=\frac{\beta}{(1-\|z\|)^{n+1}}$, $\beta>1$, if and only if $f$ is of bounded index.
Keywords	analytic function; index boundedness; composition of functions
DOI	doi:10.15330/ms.47.2.207-210
Reference	1. Sheremeta M.M. Analytic functions of bounded index. - Lviv: VNTL Publishes, 1999. 2. Goldberg A.A., Sheremeta M.N. Existence of an entire transcendental function of bounded l-index// Mat. Zametki. - 1995. - V.57, no.1 - P. 125-129 (in Russian); Engl. transl. Math Notes. - 1995. - V.57, no.1. - P. 88-90. 3. Sheremeta M.M., Bordulyak M.T. On the existence of entire functions of bounded l-index and l-regular growth// Ukrainian Math. Zh. - 1996. - V.48, no.9. - P. 1166-1182 (in Russian); Engl. transl.: Ukrainian Math. J. - 1996. - V.48, no.9. - P. 1322-1340. 4. Chyzhykov I.E., Sheremeta M.M. Boundedness of l-index for entire functions of zero genus// Mat. Stud. - 2001. - V.16, no.2. - P. 124-130. 5. Sheremeta M.M., Bordulyak M.T. Boundedness of the l-index of Laguerre-Polya entire functions// Ukrainian Math. Zh. - 2003. - V.55, no.1. - P. 91-99 (in Ukrainian); Engl. transl.: Ukrainian Math. J. - 2003. - V.55, no.1, 112-125. 6. Chyzhykov I.E., Sheremeta M.M. On the boundedness of l-index for entire functions of zero genus// Dopovidi NAN Ukraine. - 2003. - V.7. - P. 33-39. (in Ukrainian) 7. Goldberg A.A., Sheremeta M.M. On the boundedness of the l-index of canonical products// Ukr. Mat. Visn. - 2005. - V.2, no.1. - P. 52-64 (in Ukrainian); Eng. transl.: Ukr. Math. Bull. - 2005. - V.2, no.1. - P. 53-65. 8. Trukhan Yu.S., Sheremeta M.M. Boundedness of l-index of Blaschke product// Mat. Stud. - 2002. - V.17, no.2. - P. 127-137. (in Ukrainian) 9. Trukhan Yu.S., Sheremeta M.M. On l-index boundedness of the Blaschke product// Mat. Stud. - 2003. - V.19, no.1. - P. 106-112. 10. Sheremeta M.M., Trukhan Yu.S. Boundedness of the l-index of the Naftalevich-Tsuji product// Ukrainian Math. Zh. - 2004. - V.56, no.2. - P. 247-256 (in Ukrainian); Engl. transl.: Ukrainian Math. J. - 2004. - V.56, no.2. - P. 305-317. 11. Sheremeta M.N. Entire functions and Dirichlet series of bounded l-index// Izv. Vyssh. Uchebn. Zaved. Mat. - 1992. - V.9. - P. 81-87. (in Russian). Engl. transl.: Russian Math. (Iz. VUZ). - 1992. - V.36, no.9. - P. 76-82. 12. Strochyk S.M., Sheremeta M.M. Analytic in the unit disc functions of bounded index// Dopov. Akad. Nauk Ukr. - 1993. - no.1. - P. 19-22. (in Ukrainian) 13. Shah S.M. Univalence of a function f and its successive derivative when f satisfies a differential equation, II// J. Math. Anal. and Appl. - 1989. - V.142. - P. 422-430. 14. Hayman W.K. Differential inequalities and local valency// Pacific J. Math. - 1973. - V.44. - P. 117-137. 15. Sheremeta Z.M. Properties of derivatives of an entire solution of a differential equation// Mat. method. and fiz.-mech. polya. - 2006. - V.49, no.2. - P. 80-85. (in Ukrainian) 16. Sheremeta Z.M., Sheremeta M.M. Properties of entire solutions of differential equations// Ukrainian Math. Zh. - 2006. - V.58, no.12. - P. 1693-1703 (in Ukrainian); Engl. transl.: Ukrainian Math. J. - 2006. - V.56, no.12. - P. 1924-1934. 17. Sheremeta Z.M., Sheremeta M.M. On the l-index boundedness of entire solutions of a differential equation // Dopovidi NAN Ukraine. - 2007. - V.56, no.2. - P. 31-36. (in Ukrainian)
Pages	207-210
Volume	47
Issue	2
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html