A new class of functions of bounded index

Authors:
S. M. Shah and S. N. Shah

Journal:
Trans. Amer. Math. Soc. **173** (1972), 363-377

MSC:
Primary 30A66

MathSciNet review:
0313506

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Abstract: Entire functions of strongly bounded index have been defined and it is shown that functions of genus zero and having all negative zeros satisfying a one sided growth condition belong to this class.

**[1]**Ralph Philip Boas Jr.,*Entire functions*, Academic Press Inc., New York, 1954. MR**0068627****[2]**E. C. Francis and J. E. Littlewood,*Examples in infinite series with solutions*, Deighton, Cambridge, 1953.**[3]**Fred Gross,*Entire functions of bounded index*, Proc. Amer. Math. Soc.**18**(1967), 974–980. MR**0218564**, 10.1090/S0002-9939-1967-0218564-0**[4]**G. H. Hardy, J. E. Littlewood and G. Pólya,*Inequalities*, Cambridge Univ. Press, New York, 1964.**[5]**Boo-sang Lee and S. M. Shah,*On the growth of entire functions of bounded index*, Indiana Univ. Math. J.**20**(1970/1971), 81–87. MR**0259124****[6]**Benjamin Lepson,*Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index*, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., LaJolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 298–307. MR**0237788****[7]**W. J. Pugh and S. M. Shah,*On the growth of entire functions of bounded index*, Pacific J. Math.**33**(1970), 191–201. MR**0259125****[8]**S. M. Shah,*Entire functions of bounded index*, Proc. Amer. Math. Soc.**19**(1968), 1017–1022. MR**0237789**, 10.1090/S0002-9939-1968-0237789-2**[9]**S. M. Shah,*Entire functions satisfying a linear differential equation*, J. Math. Mech.**18**(1968/1969), 131–136. MR**0227410****[10]**S. M. Shah,*On entire functions of bounded index whose derivatives are of unbounded index*, J. London Math. Soc. (2)**4**(1971), 127–139. MR**0322171**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1972-0313506-8

Keywords:
Order and lower order of an entire function,
exponential type,
bounded index

Article copyright:
© Copyright 1972
American Mathematical Society