Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions

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1 International Journal of Computational Science and Mathematics. ISSN Volume, Number (00), pp International Research Publication House Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions T. Ram Reddy, R.B. Sharma and K. Saroja* Department of Mathematics, Kaatiya University, Warangal, Andhra Pradesh , India. reddytr@yahoo.com, rbsharma_005@yahoo.co.in *sarojaasula@yahoo.com Abstract The object of this paper is to introduce a new subclass of -uniformly convex functions and to obtain the coefficient inequalities, Feete-Sego inequality for the functions in this class. 000 Mathematic subject classification: Primary 0C45. Keywords: Analytic function, -uniformly convex functions, coefficient inequalities, Feete-Sego inequality. Introduction Let A denote the class of all analytic functions f of the form n f ( ) an (.) n in the open unit disc U { / C and < } Let S be the subclass of A consisting of univalent functions. Let S *( and C ( be the subclasses of S consisting of the functions f of the form (.) and satisfying the condition ' Re f > f ( ) and f "( ) Re > f '( )

2 68 T. Ram Reddy, R.B. Sharma and K. Saroja respectively for some ( 0 < ) and U. These classes are nown as starlie and convex functions of order respectively. S. Shams, S.R. Kularni and J.M.Jahangiri [4] have introduced the classes SD(, and KD (, as follows. A function f A and satisfies the condition f '( ) f '( ) Re >, U f ( ) f ( ) SD, for some 0 and 0 <. is said to be in the class Similarly KD (, be the subclass of A consisting of functions f A and satisfying the condition. f "( ) f "( ) Re >, U for some 0 and 0 <. f '( ) f '( ) For 0, these classes have been studied by S.Owa, Y.Polatoglu and E.Yavu []. They have obtained the coefficient inequalities and distortion properties for the functions in these classes. Several authors have obtained the Feete-Sego inequality for the normalied analytical functions f of the form (.) in various subclasses of S. In the present paper we introduce a new subclass of analytic functions and obtain the coefficient inequalities, Feete-Sego inequalities for the functions in this class. The results of this paper will unify and generalie the earlier results of several authors in this direction. λ be the class of functions f A satisfying the condition F () F ( ) Re > (.) F( ) F( ) where 0 α λ and 0 < with F( ) λα f ( ) ( λ α ) f ( ) ( λ α ) f ( ) It is noted that * (i) U( 0, 0,, 0) S ( and U, ( 0,, 0) C( are usual starlie functions of order and convex functions of order U, 0,, KD, Studied by Defination.: Let U (, α,, ) (ii) U ( 0, 0,, ) SD (, ) and S.Owa, Y, Polatoglu, E. Yavu [] To prove our results, we require the following Lemmas. Lemma (.) []: If p ( ) c c... is a function with positive real part then for any complex number v, we have

3 Coefficient Inequalities for a New Subclass 69 c v c max {, v } This result is sharp for the functions p( ) and p( ) Lemma (.) []: If p ( ) c c... is an analytic function with positive real part in U then for any real number v, we have c 4 v if v 0 if 0 v 4v if v vc when v < 0 or > rotations. If 0 < v the equality holds if and only if p ( ) < v then the equality holds if and only if rotations. If v 0 then the equality holds if and only if or one of its p or one of its λ λ p ( ), ( 0 λ ) or one of its rotations. If v the equality holds only for the reciprocal of p ( ) for the case v 0. Also the above upper bound is sharp and it can be further improved as follows when 0 < v <. c v c v c, 0 v / ( v) c, ( / < v ) v c c In the next sections, we obtain the coefficient inequalities and Feete-Sego in U λ, α,,. equality for the function f in the class Coefficient inequalities Theorem.: If f ( ) U ( λ, α,, ) with 0 then f ( ) U λ, α,, 0 Proof: Since f ( ) U ( λ, α,, ) F ( ) F () Re > F() F( ) F ( ) F ( ) Re > Re F() F( ) F ( ) Re > ( U ) F( ) f ( ) U λ, α,, 0 from definition (.) we have This completes the proof of the theorem (.)

4 70 T. Ram Reddy, R.B. Sharma and K. Saroja Theorem.: If f ( ) U ( λ, α,, ) ( ( )( λα ) ( an ( ) [( n )( nλα ) ] then a (.) ( n j j Proof: Since f ( ) U ( λ, α,, ) F ( ) Re > F( ), from equation (.) we have for n, U (.) Define a function p() such that F () ( ) () F n p cn, U (.4) n Here p() is analytic in U with p(0) and Re ( p ( ) ) > 0. From (.4) we get n ( ) cn [ F( ) ] ( ) F ( ) ( ) F( ) (.5) n Replacing F (), F () with their equivalent expressions in series on bothsides of the above equation (.5) we get n n ( cn [ ( n )( nλα ) ] an n n n n [( n )( nλα ) ] a c n n n n ( ) n[ ( n )( nλα ) ] a n n n ( ) ( n )( nλα ) an n Comparing the coefficient of n, on both sides of the above equation, we get ( an [ cn ( λα ) a cn n n nλα [ ] [ ( λα ) ] acn... [( n ) [( n ) λα ] ] an c [( n ) [( n ) ] ] a c ] λα n.. (.6) Taing Modulus on both sides of equation (.6) and applying Caratheodary inequality c n, we get, a n n ( [ ( λα ) a ( )( n ) [( n )( nλα ) ] n

5 Coefficient Inequalities for a New Subclass 7 [ ( λα ) ] a...[( n ) [( n ) λα ] ] an [( n ) [( n ) ] ] an ] ( When n a ( )( λα ) λα (.7) which proves the result in (.) ( ( For n [ ] a. λα Thus the result in (.) is true for n. Suppose that the result in (.) is true for 4 n m. For n m, consider a ( ( [ [ ] ] m m. m m λα ( ) ( ( ( m (... π j j ( ) [ [ ] ] m am m. m m λα j j Hence the result in (.) is true for n m. Therefore by Mathematical induction the result in (.) is true for all n. This completes the proof of the theorem (.) Feete-Sego inequality Theorem.: If f ( ) U ( λ, α,, ) ( a a ( ) [ ( λα ) ] μ ( λα ) where ( λα ), then for any complex number μ, we have {, v } μ max (.) [ ] v and the result is sharp Proof: Since f ( ) U ( λ, α,, ) from equation (.6) we have a ( )( c λα ) a c ( λα ) ac λα [ ] [ ] For any complex number μ, we have a μ a c λα λα [ ac ] [ ]

6 7 T. Ram Reddy, R.B. Sharma and K. Saroja μ ( ( ) [ λα ] c [ ] [ ] c v c a μ a (.) ( ) ( λα ) μ( ( λα ) ) where v ( λα ) Taing modulus on both sides of (.) and applying the Lemma [.], we get ( a μa max {, v } ( ) [ ( λα ) ] a μa max ( )[ ( λα ) ] ( μ[ ( λα ) ], ( λα ) This completes the proof of the theorem. And the result is sharp. a μa, if P( ) and ( ) [ ( λα ) ] ( μ[ ( λα ) ] a μa ( ) [ ( λα ) ], λα if P( ). Theorem (.):- If f U( λα,,, ) ( λα λ α ) σ ( λα λ α ) ( )( λα λ α ) σ ( ) ( λα λ α) and ( )( λα λ α ) σ 4( ) ( λα λ α) then a μa, then for any real number μ and for ( ( ) 8μ λα λ α 4 ( ) ( λα λ α ) λα λ α if μ σ

7 Coefficient Inequalities for a New Subclass 7 ( ) ( λα λ α ) if σ μ σ ( 8μ( λα λ α) 4 ( ) ( λα λ α ) ( ) ( λα λ α ) if μ σ (.) Further more if σ μ σ, then ( λα λ α ) a μa μ a, ( λα λ α ) ( ) ( λα λ α ) (.4) And if σ μ σ, then ( ( λα λ α ) ) a a μa μ ( ) ( λα λ α ) ( ) ( λα λ α ) ( ) ( ) ( λα λ α ) (.5) Proof:- Since f U( λα,,, ) a μ a where v from equation (.), we have [ ] [ ] c v c ( ) ( λα ) μ( ( λα ) ) ( λα ) Taing modulus on both sides of the above equation and applying Lemma (.) on the right hand side, we get the following cases : Case ) If μ σ, then ( λα λ α ) ( λα λ α ) μ, which on simplification, we get, v 0. c vc 4v. 8μ( λα λ α) c vc 4 ( λα λ α ). (.6) Case ) If σ μ σ, then,

8 74 T. Ram Reddy, R.B. Sharma and K. Saroja ( ) λα λ α λα λ α μ λα λ α λα λ α After simplification, we get, 0v. c vc. (.7) Case ) If μ σ, then, ( )( λα λ α ) ( ) ( λα λ α) μ after simplification, we get, v. c vc v. 4, ( λα λ α ) 8μ λα λ α c vc 4. (.8) From equations (.), (.6), (.7) and (.8), we get the result in (.) Case 4) Further more if σ μ σ, then, ( ) λα λ α λα λ α μ λα λ α 4 λα λ α After simplification, we get, 0 v. c vc v c. ( λα λ α ) ( λα λ α ) ( ) ( λα λ α ) a μa μ a Which prove the result in (.4) Case 5) Further more if σ μ σ, then, ( ) ( ) λα λ α λα λ α μ 4 λα λ α λα λ α After simplification, we get, v. c vc v c.

9 Coefficient Inequalities for a New Subclass 75 a μa ( ( ( λα λ α ) ) a μ ( ) ( λα λ α ) ( ) ( λα λ α ) ( ( ) ( λα λ α ) Which prove the result in (.5) This completes the proof of the Theorem. References [] Ma. W and Minda D.: A unified treatment of some special class of univalent functions in proceedings of the conference on complex analysis (Tianjin 99) Z. Li. F. Ren, L. Yang and S. Zhang Eds Conf. Proc. Lecture notes Anal pp International Press, Cambridge Mass, USA, 994. [] Owa S., Polatoglu Y. and Yuva E.:- Coefficient Inequalities for classes of uniformly starlie and convex functions. JIPAM volume 7 (006), Issue 5 Article 6. [] Ravichandran V., Polatoglu Y., Bolcol M. and Sen A.:-, Certain subclasses of starlie and convex functions of complex order. Haceptepe Journal of Mathematics and statistics, 4 (005), 9-5. [4] Robertson M.S.:- On the theory of univalent functions, Ann. Math.7 (96), [5] Shamas S., Kularni S.R. and Jahangiri J.M.:- Classes of uniformly starlie and convex functions Int. J. Math, Math. Sci., 55, (004),

10 76 T. Ram Reddy, R.B. Sharma and K. Saroja