On entire functions of fast growth
HTML articles powered by AMS MathViewer
- by S. K. Bajpai, G. P. Kapoor and O. P. Juneja PDF
- Trans. Amer. Math. Soc. 203 (1975), 275-297 Request permission
Abstract:
Let \[ (\ast )\quad f(z) = \sum \limits _{n = 0}^\infty {{a_n}{z^{{\lambda _n}}}} \] be a transcendental entire function. Set \[ M(r) = \max _{|z| = r} |f(z)|,\; m(r) = \max _{n \geq 0} \{ |{a_n}|{r^{{\lambda _n}}}\} \] and \[ N(r) = \max _{n \geq 0} \{ {\lambda _n}|m(r) = |{a_n}|{r^{{\lambda _n}}}\} .\] Sato introduced the notion of growth constants, referred in the present paper as ${S_q}$-order $\lambda$ and ${S_q}$-type $T$, which are generalizations of concepts of classical order and type by defining \[ (\ast \ast )\quad \lambda = \lim _{r \to \infty } \sup ({\log ^{[q]}}M(r)|\log r)\] and if $0 < \lambda < \infty$, then \[ (\ast \ast \ast )\quad T = \lim _{r \to \infty } \sup ({\log ^{[q - 1]}}M(r)|{r^\lambda })\] for $q = 2,3,4, \cdots$ where ${\log ^{[0]}}x = x$ and ${\log ^{[q]}}x = \log ({\log ^{[q - 1]}}x)$. Sato has also obtained the coefficient equivalents of $(\ast \ast )$ and $(\ast \ast \ast )$ for the entire function $(\ast )$ when ${\lambda _n} = n$. It is noted that Satoβs coefficient equivalents of $\lambda$, and $T$ also hold true for $(\ast )$ if $n$βs are replaced by ${\lambda _n}$βs in his coefficient equivalents. Analogous to $(\ast \ast )$ and $(\ast \ast \ast )$ lower ${S_q}$-order $v$ and lower ${S_q}$-type $t$ for entire function $f(z)$ are introduced here by defining \[ v = \lim _{r \to \infty } \inf ({\log ^{[q]}}M(r)|\log r)\] and if $0 < \lambda < \infty$ then \[ t = \lim _{r \to \infty } \inf ({\log ^{[q - 1]}}M(r)|{r^\lambda }),\quad q = 2,3,4, \cdots .\] For the case $q = 2$, these notions are due to Whittakar and Shah respectively. For the constant $v$, two complete coefficient characterizations have been found which generalize the earlier known results. For $t$ coefficient characterization only for those entire functions for which the consecutive principal indices are asymptotic is obtained. Determination of a complete coefficient characterization of $t$ remains an open problem. Further ${S_q}$-growth and lower ${S_q}$-growth numbers for entire function $f(z)$ we defined \[ \begin {array}{*{20}{c}} \delta \\ \mu \\ \end {array} = \lim _{r \to \infty } \begin {array}{*{20}{c}} {\sup } \\ {\inf } \\ \end {array} ({\log ^{[q - 1]}}N(r)|{r^\lambda }),\] for $q = 2,3,4, \cdots$ and $0 < \lambda < \infty$. Earlier results of Juneja giving the coefficients characterization of $\delta$ and $\mu$ are extended and generalized. A new decomposition theorem for entire functions of ${S_q}$-regular growth but not of perfectly ${S_q}$-regular growth has been found.References
- S. K. Bajpai, On the coefficients of entire functions of fast growth, Rev. Roumaine Math. Pures Appl. 16 (1971), 1159β1162. MR 304655
- R. C. Basinger, On the coefficients of an entire series with gaps, J. Math. Anal. Appl. 38 (1972), 790β792. MR 302913, DOI 10.1016/0022-247X(72)90084-4
- Alfred Gray and S. M. Shah, Holomorphic functions with gap power series, Math. Z. 86 (1965), 375β394. MR 199354, DOI 10.1007/BF01110809 O. P. Juneja, Some properties of entire functions, Dissertation, Indian Institute of Technology, Kanpur, 1965.
- O. P. Juneja, On the coefficients of an entire series of finite order, Arch. Math. (Basel) 21 (1970), 374β378. MR 274759, DOI 10.1007/BF01220932
- O. P. Juneja, On the coefficients of an entire series, J. Analyse Math. 24 (1971), 395β401. MR 281918, DOI 10.1007/BF02790381
- O. P. Juneja and Prem Singh, On the growth of an entire series with gaps, J. Math. Anal. Appl. 30 (1970), 330β334. MR 254242, DOI 10.1016/0022-247X(70)90165-4
- O. P. Juneja and G. P. Kapoor, On the lower order of entire functions, J. London Math. Soc. (2) 5 (1972), 310β312. MR 313505, DOI 10.1112/jlms/s2-5.2.310
- Daihachiro Sato, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc. 69 (1963), 411β414. MR 146381, DOI 10.1090/S0002-9904-1963-10951-9
- S. M. Shah, On the lower order of integral functions, Bull. Amer. Math. Soc. 52 (1946), 1046β1052. MR 18736, DOI 10.1090/S0002-9904-1946-08708-X
- S. M. Shah, On the coefficients of an entire series of finite order, J. London Math. Soc. 26 (1951), 45β46. MR 38423, DOI 10.1112/jlms/s1-26.1.45
- R. S. L. Srivastava and Prem Singh, On the $\lambda$-type of an entire function of irregular growth, Arch. Math. (Basel) 17 (1966), 342β346. MR 210894, DOI 10.1007/BF01899685 G. Valiron, Lectures on general theory of integral functions, Chelsea, New York, 1949. J. M. Whittakar, The lower order of integral functions, J. London Math. Soc. 8 (1933), 20-27.
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 203 (1975), 275-297
- MSC: Primary 30A64
- DOI: https://doi.org/10.1090/S0002-9947-1975-0372200-0
- MathSciNet review: 0372200