Convolution and coefficient problems for multivalent functions defined by subordination
| dc.contributor.author | Supramaniam, Shamani | |
| dc.date.accessioned | 2014-11-03T02:14:07Z | |
| dc.date.available | 2014-11-03T02:14:07Z | |
| dc.date.issued | 2009 | |
| dc.description | Master | en_US |
| dc.description.abstract | Let C be the complex plane and U := {z 2 C : |z| < 1} be the open unit disk in C and H(U) be the class of analytic functions defined in U. Also let A denote the class of all functions f analytic in the open unit disk U := {z 2 C : |z| < 1}, and normalized by f(0) = 0, and f0(0) = 1. A function f 2 A has the Taylor series expansion of the form f(z) = z + X1 n=2 anzn (z 2 U). Let Ap (p 2 N) be the class of all analytic functions of the form f(z) = zp + X1 n=p+1 anzn with A := A1. Consider two functions f(z) = zp + ap+1zp+1 + · · · and g(z) = zp + bp+1zp+1 + · · · in Ap. The Hadamard product (or convolution) of f and g is the function f g defined by (f g)(z) = zp + X1 n=p+1 anbnzn. In Chapter 1, the general classes of multi-valent starlike, convex, close-to-convex and quasi-convex functions are introduced. These classes provide a unified treatment to various known subclasses. Inclusion and convolution properties are derived using the methods of convex hull and differential subordination. In Chapter 2, bounds for the Fekete-Szeg¨o coefficient functional associated with the k-th root transform [f(zk)]1/k of normalized analytic functions f defined on U are derived for the following classes of functions: Rb(') := f 2 A : 1 + 1 b (f0(z) − 1) '(z) , S ( , ') := f 2 A : zf0(z) f(z) + z2f00(z) f(z) '(z) , L( , ') := ( f 2 A : zf0(z) f(z) 1 + zf00(z) f0(z) 1− '(z) ) , M( , ') := f 2 A : (1 − ) zf0(z) f(z) + 1 + zf00(z) f0(z) '(z) , where b 2 C \ {0} and 0. A similar problem is investigated for functions z/f(z) when f belongs to a certain class of functions. In Chapter 3, some subclasses of meromorphic univalent functions in the unit disk U are extended. Let U(p) denote the class of normalized univalent meromorphic functions f in U with a simple pole at z = p, p > 0. Let be a function with positive real part on U, (0) = 1, 0(0) > 0, which maps U onto a region starlike with respect to 1 and which is symmetric with respect to the real axis. The class P (p,w0, ) consists of functions f 2 U(p) meromorphic starlike with respect to w0 and satisfying − zf0(z) f(z) − w0 + p z − p − pz 1 − pz (z). The class P (p, ) consists of functions f 2 U(p) meromorphic convex and satisfying − 1 + z f00(z) f0(z) + 2p z − p − 2pz 1 − pz (z). The bounds for w0 and some initial coefficients of f in P (p,w0, ) and P (p, ) are obtained. | en_US |
| dc.identifier.uri | http://hdl.handle.net/123456789/250 | |
| dc.language.iso | en | en_US |
| dc.subject | Mathematical science | en_US |
| dc.subject | Multivalent functions | en_US |
| dc.subject | Subordination | en_US |
| dc.title | Convolution and coefficient problems for multivalent functions defined by subordination | en_US |
| dc.type | Thesis | en_US |
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