Applications of Cauchy-Schwarz inequalities in the mapping structure of linear operator

which are analytic in the open unit disk 1} |< :| { := z z C U  . The subclass of  consisting of functions of the form (Equation 1) which are also univalent in Uwas denoted by S . A function   f is said to be in k -UCV, the class of k -uniformly convex functions ) < (0   k if S  f along with the property that for every circular arc  contained in U , with centre  where k  | | , the image curve ) ( f is a convex arc (Kanas and Wisniowska, 1999). It is well known that (Kanas and Wisniowska, 1999) UCV   k f if and only if the image of the function p , where


INTRODUCTION
Let  denote the class of functions f normalized by .The subclass of  consisting of functions of the form (Equation 1) which are also univalent in U was denoted by S .A function   f is said to be in k -UCV, the class of k -uniformly convex functions along with the property that for every circular arc  contained in U , with centre (Kanas and Wisniowska, 1999).It is well known that (Kanas and Wisniowska, 1999) UCV   k f if and only if the image of the function p , where ), ( ) ( is a subset of the conic region: by the usual Alexander's relation as: . ) ( = ) ( where reduce to the classes of convex and starlike functions studied by Robertson (1936) and Silverman (1975) and for 1 = k , the aforementioned classes reduce to the classes of uniformly convex and uniformly starlike functions in U studied by Goodman (1991a; b)., which is starlike with respect to 1 and is also symmetric with respect to the real axis.For such functions  , Bansal (2011;2013) Caplinger and Causey (1973) and Padmanabhan (1979).
It is well known that the class S and many of its subclasses are not closed under the ring operation of usual addition and multiplication of functions.As such techniques of algebra from group theory, ring theory, etc., and those of functional analysis do not find ready applications in the class S .Therefore, the study of class- preserving and class-transforming operations is an interesting problem in geometric function theory.

PRELIMINARIES LEMMAS
Each of the following lemmas and the concept of Cauchy-Schwarz inequalities will be require for our investigation.

Lemma 1
Let the function f of the form (Equation 1) be a member of S or ST (de Branges, 1985).Then, the sharp estimate

Lemma 2
Let the function   f be of the form (Equation 1) (Bansal, 2013) . The result is sharp for the function: Let (Kanas and Wisniowska, 1999;2000): be the Riemann map of U onto k  where the region k  is defined as in Equation 2 and let the function f be given by Equation 1.
The estimates Equations 14 and 15 are sharp.

Lemma 4
Let the function   f be of the form (Equation 1) (Goodman, 1957)

Lemma 5
Let the function f , given by Equation 1 be a member of The use of Cauchy-Schwarz inequality, known as the Cauchy-Bunyakovsky-Schwarz inequality find a place in various areas of mathematics such as linear algebra, analysis, probability theory, vector algebra and many more.It is considered to be one of the most important inequalities in mathematics.It states that for complex parameters Motivated by Mishra and Panigrahi, 2011;Aouf et al., 2016;Bansal, 2013;Mostafa, 2009;Panigrahi and 10) to preserves and transform certain well known subclasses of univalent functions to another class.

MAIN RESULTS
Throughout the paper, we assume that .and 0 If the hypergeometric inequality Panigrahi and El-Ashwah 49

Proof
Let the function f given by Equation 1 be in the class S or ST .By Equation 10, we have The repeated applications of the relation Applying Cauchy-Schwarz inequality to the individual sums in Equation 21, we get Since the condition in Equation 18 is satisfied, using Gauss summation formula in Equation 22, we obtain Since gamma function is symmetric about the real axis, that is,.
, we have from Equation 23, . This ends the proof of Theorem 1. Putting in Theorem 1, after simplification we get the following result due to Mishra and Panigrahi (2011): satisfy (Mishra and Panigrahi, 2011), Theorem 1, p. 55) in Theorem 1 gives:

  f
given by Equation 1 be in class ST  k . In view of Lemma 2, it is sufficient to show that Using the coefficient estimate of Equation 14 and the elementary inequality , it is again sufficient to show that


Since the condition in Equation 24 holds, the aforementioned summation can be written as evaluation of generalized hypergeometric functions and we get Therefore, in view of Equation 26, if the hypergeometric inequality (Equation 25) is satisfied, then Taking a b = in Theorem 2, we have the following.

Let
be defined by Equation 13and C  c satisfy the inequality (Mishra and Panigrahi, 2011, Corollary 4, p. 59 in Corollary 4. Using summation formula (Equation 7), we have Hence, the result follows.
be defined by Equation 13and

Proof
The proof follows the same line to that of Theorem 2. In this case we use Lemma 4 instead of Lemma 2. The proof of Theorem 4 is complete.Takiing be defined by Equation 13and

Proof
Let the function f given by Equation 1 be a member of UCV  k . The proof follows the same line to that of Theorem 1. Making use of Lemma 2, the coefficient estimate (Equation 15) for n a and the elementary inequality , it is sufficient to show that: The term 3 S can be equivalently written as Since the condition (Equation 13) holds which ensure that sum in the r.h.s of Equation 33 are convergent hypergeometric series.Therefore, Hence, in view of Equation 32, if the hypergeometric inequality (Equation 31) is satisfied, then

Proof
Let the function f given by (1.1) be in the class given by Equation 38 ensure that the sum in the r.h.s of Equation 19 are convergent hypergeometric series so that  39) is satisfied, then This ends the proof of Theorem 7.

Conclusion
By making use of Cauchy-Schwarz inequalities, the authors obtain sufficient conditions for a linear operator define by means of normalized hypergeometric function to be certain close to convex class.In this direction, researchers (Bansal, 2013;Mostafa, 2009;Sharma et al., 2013;Sivasubramanian et al., 2011;Swaminathan, 2010;Sudharsan et al., 2014;Sivasubramanian et al., 2013) have already obtained sufficient conditions for vaious class without making use of Cauchy-Schwarz inequlaities.
Theorem 5 we get the desire result.
34 ensure that the sum in the r.h.s of Equation 37 are convergent hypergeometric series so that view of Equation 40 if the inequalities (Equation