Mapping Properties of Some Classes of Analytic Functions under Certain Integral Operators

We consider certain subclasses of analytic functions with bounded radius and bounded boundary rotation and study the mapping properties of these classes under certain integral operators.

Multivalent and Meromorphic Functions of Bounded Boundary Rotation

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-024-9 ◽

1975 ◽

Vol 27 (1) ◽

pp. 186-199 ◽

Cited By ~ 5

Author(s):

Ronald J. Leach

Keyword(s):

Critical Points ◽

Analytic Functions ◽

Unit Disc ◽

Meromorphic Functions ◽

The classVk(p). We generalize the class Vk of analytic functions of bounded boundary rotation [8] by allowing critical points in the unit disc U.Definition. Let f(z) = aqzq + . . . (q 1) be analytic in U. Then f(z) belongs to the class Vk(p) if for r sufficiently close to 1,andWe note that (1.1) implies that / has precisely p — 1 critical points in U.

On subclasses of close-to-convex functions of higher order

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s016117129200036x ◽

1992 ◽

Vol 15 (2) ◽

pp. 279-289 ◽

Cited By ~ 32

Author(s):

Khalida Inayat Noor

Keyword(s):

Convex Functions ◽

Analytic Functions ◽

Higher Order ◽

Arc Length ◽

Covering Theorem ◽

The classesTk(ρ),0≤ρ<1,k≥2, of analytic functions, using the classVk(ρ)of functions of bounded boundary rotation, are defined and it is shown that the functions in these classes are close-to-convex of higher order. Covering theorem, arc-length result and some radii problems are solved. We also discuss some properties of the classVk(ρ)including distortion and coefficient results.

Bounds on the curvature for functions with bounded boundary rotation of order1−b

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s016117128600008x ◽

1986 ◽

Vol 9 (1) ◽

pp. 65-70

Author(s):

M. A. Nasr

Keyword(s):

Analytic Functions ◽

Sharp Bounds ◽

LetVk(1−b),k≥2andb≠0real, denotes the class of locally univalent analytic functionsf(z)=z+∑n=2∞anzninD={z:|z|<1}such that∫02π|Re{1+1bzf″(z)f′(z)}|dθ<πk,z=reiθ∈D. In this note sharp bounds on the curvature of the image of|z|=r,0<r<1, under a mappingfbelonging to the classVk(1−b)have been obtained.

Coefficients of Functions with Bounded Boundary Rotation

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-161-2 ◽

1969 ◽

Vol 21 ◽

pp. 1477-1482 ◽

Cited By ~ 9

Author(s):

M. S. Robertson

Keyword(s):

Analytic Functions ◽

Bieberbach Conjecture ◽

Boundary Rotation ◽

Convex In One Direction ◽

Image Position

For fixed k ≧ 2, let Vk denote the class of normalized analytic functionssuch that z ∈ E = {z; |z| <1} are regular and have f′(0) = l,f′(z) ≠ 0, and1Let Sk be the subclass of Vk whose members f(z) are univalent in E. It was pointed out by Paatero (4) that Vk coincides with Sk whenever 2 ≦ k ≦ 4. Later Rényi (5) showed that in this case, f(z) ∈ Sk is also convex in one direction in E. In (6) I showed that the Bieberbach conjectureholds for functions convex in one direction.

Some applications of certain integral operators involving functions

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.49.2018.2369 ◽

2018 ◽

Vol 49 (1) ◽

pp. 25-34

Author(s):

Khhalida Inayat Noor ◽

Bushra Malik ◽

Syed Zakar Hussain Bukhari

Keyword(s):

Integral Transform ◽

Integral Operators ◽

Boundary Curve ◽

Integral Transforms ◽

Simply Connected Domain ◽

Boundary Rotation ◽

Bounded Radius Rotation ◽

Complete Circuit ◽

Simply Connected

Integral transforms map equations from their original domains into others where manipulations and solutions may be much easier than in original domains. To get back in the original environment, we use the idea of inverse of the integral transform. A function analytic and locally univalent in a given simply connected domain is said to be of bounded boundary rotation if its range has bounded boundary rotation which is defined as the total variation of the direction angle of the tangent to the boundary curve under a complete circuit. \qquad The main objective of the present article is to study some applications of certain integral operators to functions of bounded radius rotation involving Janowski functions. We discuss some inclusion results under certain assumption on parameters involve in operators as well as in related subclasses of analytic functions. Most of these results are best possible. We also relate our findings with the existing literature of the subjects.

On analytic functions of bounded boundary rotation of complex order

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2011.06.059 ◽

2011 ◽

Vol 62 (4) ◽

pp. 2112-2125 ◽

Cited By ~ 5

Author(s):

Khalida Inayat Noor ◽

Muhammad Aslam Noor ◽

Eisa Al-Said

Keyword(s):

Analytic Functions ◽

Complex Order ◽

Some radius of convexity problems for certain classes of analytic functions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171284000740 ◽

1984 ◽

Vol 7 (4) ◽

pp. 713-718

Author(s):

Khalida I. Noor ◽

Hailah Al-Madifer

Keyword(s):

Analytic Functions ◽

Radius Of Convexity ◽

In this paper we consider some radius of convexity problems for certain classes of analytic functions. These classes, in general, are related with functions of bounded boundary rotation.