Main Article Content



Let $K[C,D,p, \alpha]$, $- 1 \le D <C \le 1$ and $0\le \alpha <p$ denote the class of functions

\[ g(z) =z^p+\sum_{n=p+1}^\infty b_nz^n \]

analytic in the unit disc $U =\{z:|z|<1\}$ and satisfying the condition $1+\frac{zg''(z)}{g'(z)}$ is subcoordinate to $\frac{p+[pD+(C-D)(p-\alpha)]z}{1+Dz}$. We investigate the subclass of p-valent close-to-convex functions

\[ f(z) =z^p+\sum_{n=p+1}^\infty a_nz^n, \]

for which there exists $g(z)\in K[C,D,p, \alpha]$ such that $\frac{pf'(z)}{g'(z)}$ is subcoordinate to $\frac{p+[pB+(A-B)(p-\beta)]z}{1+Bz}$, $- 1 \le B <A \le 1$ and $0\le \beta <p$ . Distortion and rotation theorems and coefficient bounds are obtained.

Article Details

How to Cite
AOUF, M. K. (1991). ON SUBCLASSES OF P-VALENT CLOSE-TO-CONVEX FUNCTIONS. Tamkang Journal of Mathematics, 22(2), 133–143. https://doi.org/10.5556/j.tkjm.22.1991.4586


M. K. Aouf, "On a class of p-valent starlike functions of order alpha," International J. Math. Sci. 10(1987), no. 4, 733-744.

M. K. Aouf, "On a class of p-valent close-to-convex functions of order /3and type o," International J. Math. Math. Sci. 11(1988), no. 2, 259-266.

R. M. Goel and B. S. Mehrok, "On the coefficients of a subclass of starlike functions," Indian J. Pure Appl. Math., 12(1981), 634-637.

R. M. Goel and B. S. Mehrok, "On a class of close-to-convex functions", Indian J. Pure Appl. Math., 12(1981), 648-658.

R. M. Goel and B. S. Mehrok, "Some invariance properties of a subclass of close-to-convex functions," Indian J. Pure Appl. Math., 12(1981), 1240-1249.

E. G. Goluzian, "On the coefficients of a class of functions regular in a dis.k and having an integral representation in it", J. of Soviet Math. (2) 6, (1974), 60&-617;

W. Janowski, "Some extremal problems for certain families of analytic functions", Ann. Polon. Math., 28(1973), 297-326.

W. Kaplan, "Close-to-convex Schlicht functions," Mich. Math. J., 1(1952), 169-185.

F. R. Keogh and E. P. Merkes, "A coefficient inequality for certain classes of analytic functions," Proc. Amer. Math. Sco., 20(1969), 8-12.

J. Krzyz, "On the derivative of close-to-convex functions," Coll. Math., 10(1963), 139-142.

R. J. Libera, "Some radius of convexity problems," Duke Math. J. 31(1964), 143-158.

Z. Nehari, "Conformal Mapping", McGraw-Hill Book Co., Inc., New York (1952).

R. Mazur, "On _a subclass of convex functions," Zeszyty Nauk. Politech. todz. Mat. 13(1981), 15-20.

S. Ogawa, "A note on dose-to-convex functions," J. Nara Gakugei Univ., 8(1959), 9-10.

M. S. Robertson, "On the theory of univalent functions," Ann. of Math., 37 (1936), 169-185.

St. Ruscheweyh, A. Subordination theorem for phi-like functions," J. London Math. Soc., 13(1976), 275-280.

St. Ruscheweyh and T. Shell-Small, "Hadamard product of schilcht functions and the Polya- Schoenberg conjecture," Comment. Math. Helv., 48(1973), 119-135.

H. Silverman, "Convexity theorems for subclasses of univalent functions," Pacific J. of Maths., vol. 64, No. 1(1976), 253-263.

E. M. Silvia, "Subclasses of close-to-convex functions," Internat. J. Math. Math. Sci. Vol. 6, No. 3 (1983), 449-458.