Mapping Properties of Generalized Robertson Functions under Certain Integral Operators ()

Muhammad Arif, Wasim Ul-Haq, Muhammad Ismail

**DOI: **10.4236/am.2012.31009
PDF
HTML XML
6,181
Downloads
9,954
Views
Citations

In the present article, certain classes of generalized p-valent Robertson functions are considered. Mapping properties of these classes are investigated under certain p-valent integral operators introduced by Frasin recently.

Share and Cite:

Arif, M. , Ul-Haq, W. and Ismail, M. (2012) Mapping Properties of Generalized Robertson Functions under Certain Integral Operators. *Applied Mathematics*, **3**, 52-55. doi: 10.4236/am.2012.31009.

1. Introduction

Let be the class of functions of the form

which are analytic in the open unit disc. We write. A function is said to be spiral-like if there exists a real number

such that

The class of all spiral-like functions was introduced by L. Spacek [1] in 1933 and we denote it by. Later in 1969, Robertson [2] considered the class of analytic functions in for which.

Let be the class of functions analytic in with and

where, and is real with.

For, , this class was introduced in [3] and for, see [4]. For, and, the class reduces to the class of functions analytic in with and whose real part is positive.

We define the following classes

For, and, we obtain the well known classes and of analytic functions with bounded radius and bounded boundary rotations studied by Tammi [5] and Paatero [6] respectively. For details see [7-12]. Also it can easily be seen that and

Let us consider the integral operators

(1.1)

and

(1.2)

where and for all .

These operators, given by (1.1) and (1.2), are defined by Frasin [13]. If we take, we obtain the integral operators and introduced and studied by Breaz and Breaz [14] and Breaz et al. [15], for details see also [16-20]. Also for, in (1.1), we obtain the integral operator studied in [21] given as

and for, , in (1.2), we obtain the integral operator

discussed in [22,23].

In this paper, we investigate some propeties of the above integral operators and for the classes and respectively.

2. Main Result

Theorem 2.1. Let for with

. Also let is real with, ,

. If

then with

(2.1)

Proof. From (1.1), we have

(2.2)

or, equivalently

(2.3)

Subtracting and adding on the right hand side of (2.3), we have

(2.4)

Taking real part of (2.4) and then simple computation gives

(2.5)

where is given by (2.1). Since for, we have

(2.6)

Using (2.6) and (2.1) in (2.5), we obtain

Hence with is given by (2.1).

By setting and in Theorem 2.1, we obtain the following result proved in [9].

Corollory 2.2. Let for with. Also let,. If

then and is given by (2.1).

Now if we take and in Theorem 2.1, we obtain the following result.

Corollory 2.3. Let for with. Also let,. If

then and is given by (2.1).

Letting, , and in Theorem 2.1, we have.

Corollory 2.4. Let with. Also let. If

then

with.

Theorem 2.5. Let for

with. Also let is real is real with,

,. If

then and is given by (2.1).

Proof. From (1.2), we have

or, equivalently

This relation is equivalent to

(2.7)

Taking real part of (2.7) and then simple computation gives us

(2.8)

where is given by (2.1). Since for, we have

(2.9)

Using (2.9) in (2.8), we obtain

Hence with is given by (2.1).

By setting and in Theorem 2.5, we obtain the following result.

Corollory 2.6. Let for with. Also let,. If

then with is given by (2.1).

Letting, , and in Theorem 2.5, we have.

Corollory 2.7. Let with. Also let. If, then

with.

NOTES

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | L. Spacek, “Prispěvek k Teorii Funkei Prostych,” ?asopis pro pěstováni matematiky a fysiky, Vol. 62, No. 2, 1933, pp. 12-19. |

[2] | M. S. Robertson, “Univalent Functions f(z) for wich zf'(z) Is Spiral-Like,” Michigan Mathematical Journal, Vol. 16, No. 2, 1969, pp. 97-101. |

[3] | K. S. Padmanabhan and R. Parvatham, “Properties of a Class of Functions with Bounded Boundary Rotation,” Annales Polonici Mathematici, Vol. 31, No. 1, 1975, pp. 311-323. |

[4] | B. Pinchuk, “Functions with Bounded Boundary Rotation,” Israel Journal of Mathematics, Vol. 10, No. 1, 1971, pp. 7-16. doi:10.1007/BF02771515 |

[5] | O. Tammi, “On the Maximization of the Coefficients of Schlicht and Related Functions,” Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, Vol. 114, No. 1, 1952, 51 pages. |

[6] | V. Paatero, “Uber Gebiete von Beschrankter Randdrehung,” Annales Academiae Scientiarum Fennnicae, Vol. 37-39, No. 9, 1933. |

[7] | M. Arif, M. Ayaz and S. I. Ali Shah, “Radii Problems for Certain Classes of Analytic Functions with Fixed Second Coefficients,” World Applied Sciences Journal, Vol. 13, No. 10, 2011, pp. 2240-2243. |

[8] | K. I. Noor, “On Some Subclasses of Fuctions with Bounded Boundary and Bounded Radius Rotation,” Pan American Mathematical Journal, Vol. 6, No. 1, 1996, pp. 75-81. |

[9] | K. I. Noor, M. Arif and W. Haq, “Some Properties of Certain Integral Opertors,” Acta Universitatis Apulensis, Vol. 21, 2010, pp. 89-95. |

[10] | K. I. Noor, M. Arif and A. Muhammad, “Mapping Properties of Some Classes of Analytic Functions under an Integral Operator,” Journal of Mathematical Inequalities, Vol. 4, No. 4, 2010, pp.593-600. |

[11] | K. I. Noor, W. Haq, M. Arif and S. Mustafa, “On Bounded Boundary and Bounded Radius Rotations,” Journal of Inequalities and Applications, 2009, Article ID: 813687. |

[12] | K. I. Noor, S. N. Malik, M. Arif and M. Raza, “On Bounded Boundary and Bounded Radius Rotation Related with Janowski Function,” World Applied Sciences Journal, Vol. 12, No. 6, 2011, pp. 895-902. |

[13] | B. A. Frasin, “New General Integral Operators of p-Valent Functions,” Journal of Inequatilies Pure and Applied Mathematics, Vol. 10, No. 4, 2009 |

[14] | D. Breaz and N. Breaz, “Two Integral Operators,” Studia Universitatis Babes-Bolyai, Mathematica, Clunj-Napoca, Vol. 47, No. 3, 2002, pp. 13-21. |

[15] | D. Breaz, S. Owa and N. Breaz, “A New Integral Univalent Operator,” Acta Universitatis Apulensis, Vol. 16, 2008, pp. 11-16. |

[16] | N. Breaz, V. Pescar and D. Breaz, “Univalence Criteria for a New Integral Operator,” Mathematical and Computer Modelling, Vol. 52, No. 1-2, 2010, pp. 241-246. doi:10.1016/j.mcm.2010.02.013 |

[17] | B. A. Frasin, “Convexity of Integral Operators of p-Valent Functions,” Mathematical and Computer Modelling, Vol. 51, No. 5-6, 2010, pp. 601-605. |

[18] | B. A. Frasin, “Some Sufficient Conditions for Certain Integral Operators,” Journal of Mathematics and Inequalities, Vol. 2. No. 4, 2008, pp. 527-535. |

[19] | G. Saltik, E. Deniz and E. Kadioglu, “Two New General p-Valent Integral Operators,” Mathematical and Computer Modelling, Vol. 52, No. 9-10, 2010, pp. 1605-1609. doi:10.1016/j.mcm.2010.06.025 |

[20] | R. M. Ali and V. Ravichandran, “Integral Operators on Ma-Minda Type Starlike and Convex Functions,” Mathematical and Computer Modelling, Vo. 53, No. 5-6, 2011, pp. 581-586. doi:10.1016/j.mcm.2010.09.007 |

[21] | S. S. Miller, P. T. Mocanu and M. O. Reade, “Starlike Integral Operators,” Pacific Journal of Mathematics, Vol. 79, No. 1, 1978, pp.157-168. |

[22] | Y. J. Kim and E. P. Merkes, “On an Integral of Powers of a Spirallike Function,” Kyungpook Mathematical Journal, Vol. 12, No. 2, 1972, pp. 249-252. |

[23] | N. N. Pascu and V. Pescar, “On the Integral Operators of Kim-Merkes and Pfaltzgraff,” Mathematica, Universitatis Babes-Bolyai Cluj-Napoca, Vol. 32, No. 2, 1990, pp. 185-192. |

Journals Menu

Contact us

+1 323-425-8868 | |

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2024 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.