ISSN: 2319-8753
International Journal of Innovative Research in Science,
Engineering and Technology
(ISO 3297: 2007 Certified Organization)
Vol. 2, Issue 8, August 2013
ON A SUBCLASS OF n-UNIFORMLY MULTIVALENT
FUNCTIONS USED BY INCOMPLETE BETA
FUNCTION
Mr. Pravin Ganpat Jadhav
*Research Scholar, Shri Jagdishprasad Jhabarmal Tibrewala University, Jhunjhunu, Rajasthan
(India)
** Assistant Professor , Hon. B. J. Arts, commerce and Science College, Ale, Pune, India
ABSTRACT: In this paper we introduce the new subclasses
and
of n-uniformly
convex and n-uniformly starlike functions which are analytic and multivalent with negative coefficients. The object of this
paper is to study of the application of incomplete beta function
to n-uniformly convex multivalent functions
and obtain several properties of the subclasses
and
.
Key words and phrases : Analytic functions , Multivalent functions , Coefficient estimate, Distortion theorem, Starlike
functions , Convex functions, Close-to-close functions, n-Uniformly Convex functions, n-Uniformly Starlike functions,
Radii of close-to-close convexity , starlikeness and convexity
I. INTRODUCTION
Let
denote the class of functions
of the form
………(1)
which are analytic and multivalent in the unit disk
Let
denote the subclass of
for
consisting of functions of the form
…………….(2)
Definition 1.1 : A function
and
, denoted by
Definition 1.2 : A function
and
, denoted by
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is said to be n-uniformly starlike of order
if and only if
is said to be n-uniformly convex of order
if and only if
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Definition 1.3 : A function
and
The incomplete beta function
where
is said to be n-uniformly close to convex of order
if
is defined as
and
which is motivated from Carlson Shaffer operator defined by
Definition 1.4 : A function
is said to be in the class
if it satisfies
is said to be in the class
if and only if
Where
Definition 1.5 : A function
is in
the class
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II. COEFFICIENT ESTIMATES
THEOREM 1. A function
and defined by
is in the class
if and only if
PROOF: A function
is in the class
Therefore we have
………………………..(1.1)
Using the fact for ,
………………..(1.2)
Letting
Therefore from (1.2) we have
…………….(1.3)
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Vol. 2, Issue 8, August 2013
=
From (1.3) we have
The last inequality holds for all
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. letting
yields
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By mean value theorem we obtain
COROLLARY 1.1:-
then
COROLLARY 1.2:- for
we have
,
COROLLARY 1.3:- for
we have
THEOREM 2. A function
and defined by
is in the class
if and only if
PROOF: A function
Therefore we have
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is in the class
is in class of
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Let
=
………………………..(2.1)
Using the fact for ,
………………..(2.2)
Letting
Therefore from (1.2) we have
…………….(2.3)
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=
From (2.3) we have
The last inequality holds for all
. letting
yields
By mean value theorem we obtain
COROLLARY 2.1:-
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then
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COROLLARY 2.2:- for
we have
,
COROLLARY 2.3:- for
we have
III. GROWTH AND DISTORTION THEOREM
THEOREM 3:- If
then
With equality hold for
PROOF :-
Therefore from (1.1)
Therefore
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Similarly
Therefore
THEOREM 4:- If
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then
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With equality hold for
PROOF :-
Therefore from (2.1)
Therefore
Similarly
Therefore
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THEOREM 5:- If
then
With equality hold for
PROOF :-
Therefore from (1.1)
Therefore
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Similarly
Therefore
THEOREM 6:- If
then
With equality hold for
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PROOF :-
Therefore from (2.1)
Therefore
Similarly
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Therefore
IV. RADII OF CLOSE-TO-CONVEXITY, STARLIKENESS AND CONVEXITY
THEOREM 7:- If
, then
is close to convex of order
in
where
PROOF:- It is sufficient to show that
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………..(7.1)
from (1.1)
That is
Observe that (7.1) is true if
Therefore
, which complete the proof.
THEOREM 8:- If
, then
is starlike of order
in
where
PROOF:- We must show that
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We have
……………(8.1)
Hence (8.1) holds true if
)(
)
Or equivalently
………..(8.2)
From (1.1) we have
Hence by using (8.2) and (8.3) we get
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Vol. 2, Issue 8, August 2013
, which complete the proof.
THEOREM 9:- If
, then
is convex of order
in
where
PROOF:- We know that
is convex if and only if
is starlike
We must show that
Where
Therefore we have
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(9.1)
From (1.1) we have
Hence by using (9.1) and (9.2) we get
, which complete the proof.
V. CLOSURE THEOREM
THEOREM 10 :
Let
and
Then
for
if and only if
where
PROOF: - Suppose
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can be expressed in the form
and
can be expressed in the form
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Vol. 2, Issue 8, August 2013
Then
Therefore
Conversely, suppose that
We have
We take
and
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REFERENCES
[1]
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Jadhav P. G. and S. M. Khairnar, Certain subclasses of analytic and multivalent functions in the unit disc International J. of Math. Sci. & Engg.
Appls. (IJMSEA) Vol. 6 No. VI (November, 2012), pp.175-192
Jadhav P. G. and S. M. Khairnar , Certain family of meromorphic and multivalent functions defined by linear operator in the unit disc ,
International J. of Math. Sci. & Engg. Appls. (IJMSEA) Vol. 7 No. II (Mar, 2013), pp.15-33
Jadhav P. G. and S. M. Khairnar , Some application of differential operator to analytic and multivalent function with negative coefficient ,
International J. of Manage.Studies, Statistics & App. Economics.(IJMSAE)ISSN 2250-0367, Vol. 3 No. I (June, 2013), pp.111-128.
S. M. Khairnar and Meena More, On a Class of Meromorphic Multivalent Functions with Negative Coefficients Defined by Ruscheweyh
Derivative, International Mathematical Forum, 3, 2008, no. 22, 1087 – 1097
S. M. Khairnar and Meena More, On a subclass of multivalent -Uniformly starlike and convex functions defined by a linear operator ,
IAENG International Journal of Applied Mathematics , 39:3, IJAM_39_06 , 2009
[6]
H. M. Srivastava , T.N. Shanmugam, C. Ramachandran and S. Sivasubramanian , A new subclass of
uniformly convex functions with
negative coefficient , journal of inequalities in pure and applied mathematics , ISSN: 1443-5756 , volume 8 (2007),Issue2, Artical43, pp. 14.
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