Analytic functions with univalent derivatives and entire functions of exponential type

Shah, S. M.

doi:10.1090/S0002-9904-1972-12900-8

Analytic functions with univalent derivatives and entire functions of exponential type
HTML articles powered by AMS MathViewer

by S. M. Shah PDF

Bull. Amer. Math. Soc. 78 (1972), 154-171

References

Dov Aharonov, Proof of the Bieberbach conjecture for a certain class of univalent functions, Israel J. Math. 8 (1970), 102–104. MR 265577, DOI 10.1007/BF02771305
Jochen Becker, Über Subordinationsketten und quasikonform fortsetzbare schlichte Funktionen, Technische Universität Berlin, Berlin, 1970. Von der Fakultät für Allgemeine Ingenieurwissenschaften der Technischen Universität Berlin zur Verleihung des akademischen Grades Doktor der Naturwissenschaften genehmigte Dissertation. MR 0437751
S. D. Bernardi, A survey of the development of the theory of schlicht functions, Duke Math. J. 19 (1952), 263–287. MR 48579
S. D. Bernardi, Bibliography of Schlicht functions, Office of Naval Research, New York; New York University, New York, 1966. Courant Institute of Mathematical Sciences, New York University, Tech. Report No. NR 041-019, IMM 351. MR 0202990
S. D. Bernardi, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 24 (1970), 312–318. MR 251202, DOI 10.1090/S0002-9939-1970-0251202-X
R. P. Boas Jr., Univalent derivatives of entire functions, Duke Math. J. 6 (1940), 719–721. MR 2601
Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
Enrico Bombieri, On the local maximum property of the Koebe function, Invent. Math. 4 (1967), 26–67. MR 218549, DOI 10.1007/BF01404579
D. A. Brannan, J. Clunie, and W. E. Kirwan, Coefficient estimates for a class of star-like functions, Canadian J. Math. 22 (1970), 476–485. MR 260994, DOI 10.4153/CJM-1970-055-8
J. D. Buckholtz, The Whittaker constant and successive derivatives of entire functions, J. Approximation Theory 3 (1970), 194–212. MR 260998, DOI 10.1016/0021-9045(70)90027-4
J. D. Buckholtz, Successive derivatives of analytic functions, Indian J. Math. 13 (1971), 83–88. MR 333176
J. D. Buckholtz and J. L. Frank, Whittaker constants, Proc. London Math. Soc. (3) 23 (1971), 348–370. MR 296297, DOI 10.1112/plms/s3-23.2.348
Douglas M. Campbell, Joseph A. Cima, and John A. Pfaltzgraff, Linear spaces and linear-invariant families of locally univalent analytic functions, Manuscripta Math. 4 (1971), 1–30. MR 288294, DOI 10.1007/BF01168902
J. A. Cima and J. A. Pfaltzgraff, A Banach space of locally univalent functions, Michigan Math. J. 17 (1970), 321–334. MR 278052
W. M. Causey, The close-to-convexity and univalence of an integral, Math. Z. 99 (1967), 207–212. MR 215972, DOI 10.1007/BF01112451
W. M. Causey and E. P. Merkes, Radii of starlikeness of certain classes of analytic functions, J. Math. Anal. Appl. 31 (1970), 579–586. MR 264047, DOI 10.1016/0022-247X(70)90010-7
Z. Charzyński and M. Schiffer, A new proof of the Bieberbach conjecture for the fourth coefficient, Arch. Rational Mech. Anal. 5 (1960), 187–193 (1960). MR 114930, DOI 10.1007/BF00252902
J. Clunie, On integral functions having prescribed asymptotic growth, Canadian J. Math. 17 (1965), 396–404. MR 176075, DOI 10.4153/CJM-1965-040-8
J. Clunie, On integral functions having prescribed asymptotic growth, Canadian J. Math. 17 (1965), 396–404. MR 176075, DOI 10.4153/CJM-1965-040-8
V. F. Cowling and W. C. Royster, Domains of variability for univalent polynomials, Proc. Amer. Math. Soc. 19 (1968), 767–772. MR 227392, DOI 10.1090/S0002-9939-1968-0227392-2
Albert Edrei and Wolfgang H. J. Fuchs, Entire and meromorphic functions with asymptotically prescribed characteristic, Canadian J. Math. 17 (1965), 383–395. MR 174742, DOI 10.4153/CJM-1965-039-7
Paul Eenigenburg, On the radius of curvature for convex analytic functions, Canadian J. Math. 22 (1970), 486–491. MR 260988, DOI 10.4153/CJM-1970-056-5
P. J. Eenigenburg and F. R. Keogh, The Hardy class of some univalent functions and their derivatives, Michigan Math. J. 17 (1970), 335–346. MR 306505, DOI 10.1307/mmj/1029000519
Keith A. Ekblaw, The functions of bounded index as a subspace of a space of entire functions, Pacific J. Math. 37 (1971), 353–355. MR 302926, DOI 10.2140/pjm.1971.37.353
P. Erdös and A. Rényi, On the number of zeros of successive derivatives of analytic functions, Acta Math. Acad. Sci. Hungar. 7 (1956), 125–144 (English, with Russian summary). MR 80155, DOI 10.1007/BF02028197
G. Fricke and R. E. Powell, A theorem on entire methods of summation, Compositio Math. 22 (1970), 253–259. MR 271577
G. H. Fricke and R. E. Powell, Bounded index and summability methods, J. Austral. Math. Soc. Ser. A 21 (1976), no. 1, 79–87. MR 393924, DOI 10.1016/s0022-5193(76)80042-2
S. Friedland and Z. Nehari, Univalence conditions and Sturm-Liouville eigenvalues, Proc. Amer. Math. Soc. 24 (1970), 595–603. MR 251204, DOI 10.1090/S0002-9939-1970-0251204-3
P. R. Garabedian, G. G. Ross, and M. M. Schiffer, On the Bieberbach conjecture for even $n$, J. Math. Mech. 14 (1965), 975–989. MR 0182725
P. R. Garabedian and M. Schiffer, The local maximum theorem for the coefficients of univalent functions, Arch. Rational Mech. Anal. 26 (1967), 1–32. MR 225994, DOI 10.1007/BF00283856
G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039, DOI 10.1090/mmono/026
A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc. 8 (1957), 598–601. MR 86879, DOI 10.1090/S0002-9939-1957-0086879-9
A. W. Goodman, The valence of sums and products, Canadian J. Math. 20 (1968), 1173–1177. MR 231988, DOI 10.4153/CJM-1968-111-0
A. W. Goodman, Open problems on univalent and multivalent functions, Bull. Amer. Math. Soc. 74 (1968), 1035–1050. MR 231989, DOI 10.1090/S0002-9904-1968-12045-2
Fred Gross, Entire functions of bounded index, Proc. Amer. Math. Soc. 18 (1967), 974–980. MR 218564, DOI 10.1090/S0002-9939-1967-0218564-0
Fred Gross, Entire functions of exponential type, J. Res. Nat. Bur. Standards Sect. B 74B (1970), 55–59. MR 265592, DOI 10.6028/jres.074B.007
W. K. Hayman, Multivalent functions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 48, Cambridge University Press, Cambridge, 1958. MR 0108586
W. K. Hayman, Coefficient problems for univalent functions and related function classes, J. London Math. Soc. 40 (1965), 385–406. MR 179344, DOI 10.1112/jlms/s1-40.1.385
W. K. Hayman, Research problems in function theory, The Athlone Press [University of London], London, 1967. MR 0217268
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
J. A. Hummel, The coefficients of starlike functions, Proc. Amer. Math. Soc. 22 (1969), 311–315. MR 251209, DOI 10.1090/S0002-9939-1969-0251209-4
V. Ganapathy Iyer, On the space of integral functions. I, J. Indian Math. Soc. (N.S.) 12 (1948), 13–30. MR 27948
Z. J. Jakubowski, Sur les coefficients des fonctions univalentes dans le cercle unité, Ann. Polon. Math. 19 (1967), 207–233 (French). MR 215979, DOI 10.4064/ap-19-2-207-233
James A. Jenkins and Kôtaro Oikawa, On results of Ahlfors and Hayman, Illinois J. Math. 15 (1971), 664–671. MR 296271
Wilfred Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169–185 (1953). MR 54711
F. R. Keogh, A strengthened form of the ${1\over 4}$ theorem for starlike univalent functions, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 201–211. MR 0274736
Amy C. King and S. M. Shah, Indices of Lindelöf functions and their derivatives, Rocky Mountain J. Math. 2 (1972), no. 4, 579–593. MR 310243, DOI 10.1216/RMJ-1972-2-4-579
K. Knopp and G. G. Lorentz, Beiträge zur absoluten Limitierung, Arch. Math. 2 (1949), 10–16 (German). MR 32790, DOI 10.1007/BF02036747
J. G. Krzyż, On close-to-convex functions, Symposia on Theoretical Physics and Mathematics, Vol. 10 (Inst. Math. Sci., Madras, 1969) Plenum, New York, 1970, pp. 23–27. MR 0260990
Boo-sang Lee and S. M. Shah, The type of an entire function satisfying a linear differential equation, Arch. Math. (Basel) 20 (1969), 616–622. MR 255884, DOI 10.1007/BF01899062
Boo-sang Lee and S. M. Shah, An inequality involving the Bessel function and its derivatives, J. Math. Anal. Appl. 30 (1970), 144–156. MR 255867, DOI 10.1016/0022-247X(70)90189-7
Boo-sang Lee and S. M. Shah, Inequalities satisfied by entire functions and their derivatives, Trans. Amer. Math. Soc. 149 (1970), 109–117. MR 257355, DOI 10.1090/S0002-9947-1970-0257355-6
Boo-sang Lee and S. M. Shah, On the growth of entire functions of bounded index, Indiana Univ. Math. J. 20 (1970/71), 81–87. MR 259124, DOI 10.1512/iumj.1970.20.20006
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., La Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 298–307. MR 0237788
Norman Levinson, A theorem of Boas, Duke Math. J. 8 (1941), 181–182. MR 3802
R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755–758. MR 178131, DOI 10.1090/S0002-9939-1965-0178131-2
Karl Löwner, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann. 89 (1923), no. 1-2, 103–121 (German). MR 1512136, DOI 10.1007/BF01448091
Thomas H. MacGregor, Univalent power series whose coefficients have monotonic properties, Math. Z. 112 (1969), 222–228. MR 249596, DOI 10.1007/BF01110223
V. Marić and S. M. Shah, Entire functions defined by gap power series and satisfying a differential equation, Tohoku Math. J. (2) 21 (1969), 621–631. MR 259121, DOI 10.2748/tmj/1178242906
E. P. Merkes, M. S. Robertson, and W. T. Scott, On products of starlike functions, Proc. Amer. Math. Soc. 13 (1962), 960–964. MR 142747, DOI 10.1090/S0002-9939-1962-0142747-7
Zeev Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551. MR 29999, DOI 10.1090/S0002-9904-1949-09241-8
Zeev Nehari, Conformal mapping, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR 0045823
Mitsuru Ozawa, An elementary proof of the Bieberbach conjecture for the sixth coefficient, K\B{o}dai Math. Sem. Rep. 21 (1969), 129–132. MR 247064
K. S. Padmanabhan, On the radius of univalence of certain classes of analytic functions, J. London Math. Soc. (2) 1 (1969), 225–231. MR 247062, DOI 10.1112/jlms/s2-1.1.225
Roger N. Pederson, A proof of the Bieberbach conjecture for the sixth coefficient, Arch. Rational Mech. Anal. 31 (1968/69), 331–351. MR 239072, DOI 10.1007/BF00251415
G. Polya, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178–191. MR 7781, DOI 10.1090/S0002-9904-1943-07853-6
Christian Pommerenke, Über die Mittelwerte und Koeffizienten multivalenter Funktionen, Math. Ann. 145 (1961/62), 285–296 (German). MR 133448, DOI 10.1007/BF01451371
Walter Pugh, Sums of functions of bounded index, Proc. Amer. Math. Soc. 22 (1969), 319–323. MR 243067, DOI 10.1090/S0002-9939-1969-0243067-9
W. J. Pugh and S. M. Shah, On the growth of entire functions of bounded index, Pacific J. Math. 33 (1970), 191–201. MR 259125, DOI 10.2140/pjm.1970.33.191
G. A. Read, Univalent derivatives of entire functions, J. London Math. Soc. (2) 1 (1969), 189–192. MR 249617, DOI 10.1112/jlms/s2-1.1.189
G. A. Read, Related interpolation problems, J. London Math. Soc. (2) 2 (1970), 353–357. MR 264286, DOI 10.1112/jlms/s2-2.2.353
M. S. Robertson, Quasi-subordination and coefficient conjectures, Bull. Amer. Math. Soc. 76 (1970), 1–9. MR 251210, DOI 10.1090/S0002-9904-1970-12356-4
M. S. Robertson, The sum of univalent functions, Duke Math. J. 37 (1970), 411–419. MR 264052, DOI 10.1215/S0012-7094-70-03752-X
W. C. Royster, On the univalence of a certain integral, Michigan Math. J. 12 (1965), 385–387. MR 183866, DOI 10.1307/mmj/1028999421
S. M. Shah, Entire functions of bounded index, Proc. Amer. Math. Soc. 19 (1968), 1017–1022. MR 237789, DOI 10.1090/S0002-9939-1968-0237789-2
S. M. Shah, Entire functions satisfying a linear differential equation, J. Math. Mech. 18 (1968/1969), 131–136. MR 0227410, DOI 10.1512/iumj.1969.18.18013
S. M. Shah, Univalent derivatives of entire functions of slow growth, Arch. Rational Mech. Anal. 35 (1969), 259–266. MR 247083, DOI 10.1007/BF00248161
Swarupchand M. Shah, Entire functions of unbounded index and having simple zeros, Math. Z. 118 (1970), 193–196. MR 273015, DOI 10.1007/BF01113342
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 322171, DOI 10.1112/jlms/s2-4.1.127
S. M. Shah, Holomorphic functions with mean $p$-valent derivatives, Math. Ann. 193 (1971), 176–182. MR 293081, DOI 10.1007/BF02052388
S. M. Shah and W. C. Sisarcick, On entire functions of exponential type, J. Res. Nat. Bur. Standards Sect. B 75B (1971), 141–147. MR 310248, DOI 10.6028/jres.075B.004
S. M. Shah and S. Y. Trimble, Univalent functions with univalent derivatives, Bull. Amer. Math. Soc. 75 (1969), 153–157. MR 239074, DOI 10.1090/S0002-9904-1969-12186-5
S. M. Shah and S. Y. Trimble, Univalent functions with univalent derivatives. II, Trans. Amer. Math. Soc. 144 (1969), 313–320. MR 249598, DOI 10.1090/S0002-9947-1969-0249598-4
S. M. Shah and S. Y. Trimble, Univalent functions with univalent derivatives. III, J. Math. Mech. 19 (1969/1970), 451–460. MR 0251207, DOI 10.1512/iumj.1970.19.19042
S. M. Shah and S. Y. Trimble, Entire functions with univalent derivatives, J. Math. Anal. Appl. 33 (1971), 220–229. MR 280716, DOI 10.1016/0022-247X(71)90194-6
Vikramaditya Singh, Grunsky inequalities and coefficients of bounded schlicht functions. , Ann. Acad. Sci. Fenn. Ser. A I No. 310 (1962), 22. MR 0143894
D. C. Spencer, On mean one-valent functions, Ann. of Math. (2) 42 (1941), 614–633. MR 4903, DOI 10.2307/1969250
T. J. Suffridge, A coefficient problem for a class of univalent functions, Michigan Math. J. 16 (1969), 33–42. MR 240297, DOI 10.1307/mmj/1029000163
T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. 37 (1970), 775–777. MR 269827, DOI 10.1215/S0012-7094-70-03792-0
Donald J. Wright, The radius of $p$-valent starlikeness for certain classes of analytic functions, Ann. Polon. Math. 23 (1970/71), 49–55. MR 264056, DOI 10.4064/ap-23-1-49-55
Michael R. Ziegler, Some integrals of univalent functions, Indian J. Math. 11 (1969), 145–151. MR 259101