Coefficient inequalities for certain subclasses Of -valent functions R.B. Sharma and K. Saroja* Deartment of Mathematics, Kakatiya University, Warangal, Andhra Pradesh - 506009, India. rbsharma_005@yahoo.co.in *sarojakasula@yahoo.com ABSTRACT The aim of the resent aer is to introduce two new subclasses of -valent functions with comlex order. The coefficient inequalities and Fekete-Sego inequality for the functions in these classes are also obtained. 000 Mathematic subject classification. Primary 30C45. Key Words: -valent functions, comlex order, coefficient inequalities, Fekete-Sego inequality. * Corresonding Author. Let. Introduction A denote the class of all -valent functions f of the form f = + a n+ (.) n= n+ Which are regular in the oen unit disc U = C: Here A A and N. Let M and satisfying the conditions N be the classes consisting of the functions f A and 3
f Re < U, f and f" Re + < U, f' resectively These classes were introduced by S.Owa and H.M.Srivastava [0] Y.Polatoglu, M.Bolcol, A.Sen and E.Yavu [4] have studied the subordination results, coefficient inequalities, distortion roerties, radius of starlikeness for the functions in M. Several authors [3,4,5,7,9] have obtained the Fekete-Sego inequality for functions in various subclasses of analytic, -valent, meromorhic functions. In this aer, we define some subclasses of -valent functions of comlex order. We obtain the coefficient inequality and Fekete-Sego inequality, for the functions in these classes. Definition.: Let b be a non-ero comlex number and. A function f of the form (.) is said to be in the class M b, if f Re <, U b f (.) It is noted that M, M defined by S.Owa and H.M.Srivastava [0] 4
M, M defined by S.Owa and J.Nishiwaki [] Definition.: Let b be a non-ero comlex number and. A function f of the form (.) is said to be in the class N b, if f " Re <, U b f ' (.3) It is noted that N, N defined by S.Owa and H.M.Srivastava [0] N, N defined by S.Owa and J.Nishiwaki [] To rove our results we require the following lemma. Lemma (.) [9]: If c c... is a function with ositive real art and 0 then for any comlex number v, we have c v c Max, v This result is shar for the functions And In the next sections we obtain the coefficient inequality and Fekete-Sego inequality for the function f in the classes M b, and, N b.. Coefficient inequalities 5
Theorem.: If f M b, then n an b j n! (.) j0 Proof: Since f M b, then from the definition (.), we have ' f Re. b f Define a function such that ' f b f n cn, U (.) n b Here is a function with ositive real art with 0. Relacing f, f with their equivalent exressions on both sides, we get n n c n b b an n n n n b b an n an n n (.3) Comaring the coefficient of n on both sides of equation (.3), We get, 6
... nan b cn a cn a cn an c (.4) Taking modulus on both sides of (.4) and alying c n we get n For n = b an a a... an an n a b (.5) Thus the result holds true for n = For n = b a b Thus the result (.) is true for n =. Suose the result (.) is true for n = k Now for n = k +, we have k b ak b b b... k... b j k! j0 k ak b j k! j0 Thus the result (.) is true for nk. By mathematical induction the result (.) is true for all values of n. This comletes the roof of the theorem. Theorem.: If f N b, then 7
n an b j n! n (.6) j0 Proof: Since f N b, then from the definition (.), we have f " Re b f '. Define a function such that f " b f ' b n c n n (.7) Here is a function with ositive real art and 0. Relacing f, f & f " sides, we get with their equivalent exressions in series on both n n n n b b cn an n cn an n n n n n n n b b an n an n n n n (.8) Comaring the coefficient of n on both sides of equation (.8), We get, 8
cn a cn a cn... nn an b b.(.9) a n n c a n c n Taking modulus on both sides of (.9) and alying c n we get n a n b a a... (.0) nn n a n a n n For n = a b. Thus the result (.6) is true for n =. For n = b a b Thus the result (.6) holds true for n =. Suose the result (.6) is true for n = k Now for n = k + Consider b k k b ak b b... k... b j k! j0 k ak b j k! k Thus the result (.6) is true for nk. By mathematical induction the result (.6) is true for all values of n. This comletes the roof of the theorem. j0 9
0
Theorem 3.: If f M b, 3. Fekete Sego Inequalities then for any comlex number we have a a b b max, And the result is shar. Proof: Since f M b, then from equation (.4), we have a b c b c And a b c a c b a c b c For any comlex number we have b a a c b c b c b a a c b c b c b c b c
b a a c vc Where v b Taking modulus on both sides and by alying Lemma (.), we get b a a c vc max, b v b max, b This roves the result (3.). The result is shar. if a a b b b This comletes the roof of the theorem. if Theorem 3.: If f N b, then for any comlex number we have a a b max, b and the result is shar. Proof: If f N b, then from equation (.9), we have a b b c b c (3.5)
And b b a c a c b a c b c (3.6) For any comlex number we have b a a c b c b c c b c b a a b c b a a c vc Where v b Taking modulus on both sides and by alying Lemma (.), we get b a a c vc 3
b max, v a a b max, b This roves the result (3.4) and is shar, i.e. a a b if b b if This comletes the roof of the theorem. Acknowledgements: The authors are very much thankful to Prof.T. Ram Reddy for his valuable suggestions throughout this work. 4
References. K.Saroja:- Coefficient inequalities for some subclasses of Analytic, Univalent and Multivalent functions (00), Kakatiya University, Warangal.. Owa. S and Nishiwaki J.:- Coefficient estimates for certain classes of analytic functions, JIPAM, volume 3, Issue 5, Article 7, 00. 3. Owa.S, Polatoglu.Y, Yavu.E.:- Coefficient inequality for classes of uniformly starlike and convex function Jiam, Volume-, Issue, Article 60, 006. 4. Polataglu Y., Sen A., Bolcol B. and Yuva E.:- An investigation on a subclass of P-valently starlike functions in the unit disc. Turk, J. Math 3 (007), -8. 5. R.B.Sharma & T.Ram Reddy:- A coefficient inequality for certain subclasses of analytic functions. Indian Journal of Mathematics. The Allahabad Mathematical Society, Vol.53, No., Aril 0. 6. R.B.Sharma & T.Ram Reddy:- Coefficient inequality for certain subclasses of analytic functions. International Journal of Mathematical Analysis (Hikary Ltd), Vol.IV (00) No.3 6 ISSN 3 8876. 7. R.B.Sharma, T. Ram Reddy & K.Saroja:- Coefficient inequalities for a subclass of restarlike functions International Journal of Contemorary Mathematical Sciences, Vol.6, 0, No.45. 8. R.B.Sharma:- Fekete-Sego inequality for some sub-classes of Analytic Functions, Ph.D. Thesis, Kakatiya University, Warangal (008). 9. Ravichandran V., Polatoglu Y., Bolcol M. and Sen A.:-, Certain subclasses of starlike and convex functions of comlex order. Haccttee Journal of Mathematics and statistics, 34 (005), 9-5. 0. Srivastava H.M. and Owa.S: Some generalied convolution roerties associated with certain subclasses of analytic functions, J. Ineq. Pure Al. Math 3(3) (00). 5