Bull Korean Math Soc 43 2006, No 1, pp 179 188 A CLASS OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS DEFINED BY CONVOLUTION Rosihan M Ali, M Hussain Khan, V Ravichandran, and K G Subramanian Abstract For a given p-valent analytic function g with positive coefficients in the open unit disk, we study a class of functions fz = z p a nz n a n 0 satisfying 1 p R zf g z f gz > α 0 α < 1; z Coefficient inequalities, distortion and covering theorems, as well as closure theorems are determined The results obtained extend several known results as special cases 1 Introduction Let Ap, m be the class of all p-valent analytic functions fz = z p + a nz n defined on the open unit disk := {z C : z < 1} and let A := A1, 2 For two functions fz = z p + a nz n and gz = z p + b nz n in Ap, m, their convolution or Hadamard product is defined to be the function f gz := z p + a nb n z n Let T p, m be the subclass of Ap, m consisting of functions of the form 11 fz = z p a n z n a n 0 for n m Received December 7, 2004 2000 Mathematics Subject Classification: 30C45 Key words and phrases: starlike function, convolution, subordination, negative coefficients The authors R M Ali and V Ravichandran acknowledged support from an IRPA grant 09-02-05-00020 EAR
180 R M Ali, M H Khan, V Ravichandran, and K G Subramanian and let T := T 1, 2 A function fz T p, m is called a function with negative coefficients The subclass of T p, m consisting of multivalent starlike convex functions of order α is denoted by T S p, m, α T Cp, m, α The classes T S α := T S 1, 2, α and T Cα := T C1, 2, α were studied by Silverman [4] In this article, we study the class T Sgp, m, α introduced in the following: Definition 1 Let gz = z p + b nz n be a fixed function in Ap, m with b n > 0 n m The class T Sgp, m, α consists of functions fz of the form 11 that satisfies 12 1 p R zf g z f gz > α 0 α < 1; z Several well-known subclasses of functions are special cases of our class for suitable choices of gz when p = 1 and m = 2 For example, if gz := z/1 z, the class T Sgp, m, α is the class T S α of starlike functions with negative coefficients of order α introduced and studied by Silverman [4] If gz := z/1 z 2, the class T Sgp, m, α is the class T Cα of convex functions with negative coefficients of order α z See Silverman [4] If gz :=, λ > 1, p = 1, the class 1 z λ+1 T Sgp, m, α reduces to the class T λ α := {f T : R zdλ fz } D λ > α, z, λ > 1, α < 1, fz introduced and studied by Ahuja [1] where D λ denotes the Ruscheweyh derivatives of order λ When gz := z + n=2 nl z n, the class T Sgp, m, α is the class T Sl α where zd T Sl {f l α := fz } T : R D l > α fz Here D l denotes the Salagean derivative of order l [3] A function f Ap, m is β-pascu convex of order α if 1 1 βzf p R z + β p zzf z 1 βfz + β p zf > α β 0; 0 α < 1 z We denote by T P Cp, m, α, β the subclass of T p, m consisting of β- Pascu convex functions of order α Clearly T S α and T Cα are special cases of T P C1, 2, α, β In this paper, we obtain the coefficient inequalities, distortion and covering theorems, as well as closure theorems for functions in the class
Functions with negative coefficients 181 T Sgp, m, α Several known results are easily deduced from ours, for example, results for the classes T λ α and T Sl α Additionally, we present results for the α-pascu convex functions that unifies corresponding results for T S α and T Cα 2 The class T Sg p, m, α We first prove a necessary and sufficient condition for functions to be in T S gp, m, α in the following: Theorem 1 A function fz given by 11 is in T Sgp, m, α if and only if 21 n pαa n b n Proof If f T Sgp, m, α, then 21 follows from 12 by letting z 1 through real values To prove the converse, assume that 21 holds Then by making use of 21, we obtain zf g z pf gz f gz or f T S gp, m, α n pa nb n 1 a nb n Corollary 1 A function fz given by 11 is in T P Cp, m, α, β if and only if n pα[1 βp + βn] p 2 1 α As an immediate application of Theorem 1, we obtain the following: Theorem 2 Let fz be given by 11 If f T S gp, m, α, then with equality only for functions of the form f n z = z p z n Proof If f T S gp, m, α, then, by making use of 21, we obtain n pαa n b n n pαa n b n
182 R M Ali, M H Khan, V Ravichandran, and K G Subramanian or Clearly for f n z = z p p1 α n pαb n z n T S gp, m, α, we have a n = Corollary 2 Let fz be given by 11 If f T P Cp, m, α, β, then p 2 1 α n pα[1 βp + βn] with equality only for functions of the form f n z = z p p 2 1 α n pα[1 βp + βn] zn By making use of Theorem 1, we obtain the following growth estimate for functions in the class T S gp, m, α Theorem 3 If f T S gp, m, α, then r p r m fz r p + r m, z = r < 1, provided b n b m n m The result is sharp with equality for 22 fz = z p at z = r and z = re iπ2k+1 m p 23 k Z z m Proof Let z = r Since fz = z p a nz n, we have fz r p + a n r m Since for n m, r p + r m a n,
using 21 yields or 24 b m m pα Functions with negative coefficients 183 n pαa n b n This together with 23 shows that and similarly we have fz r p m + r fz r p m r p1 α m pαb m Let < 1 By letting r 1 in Theorem 3, we see that functions f T Sgp, m, α map the unit disk onto regions that contained the disk w < 1 p1 α m pαb m Corollary 3 If f T P Cp, m, α, β, then r p p 2 1 α m pα[1 βp + βm] rm fz r p + The result is sharp for 25 fz = z p p 2 1 α m pα[1 βp + βm] rm, z = r < 1 p 2 1 α m pα[1 βp + βm] zm We now prove the distortion theorem for the functions in T S gp, m, α in the following: Theorem 4 If f T S gp, m, α, then pr p 1 m r m 1 f z pr p 1 + m r m 1, z = r < 1,
184 R M Ali, M H Khan, V Ravichandran, and K G Subramanian provided b n b m The result is sharp for fz given by 22 Proof For a function f T Sgp, m, α, it follows from 21 and 24 that m n Since the remaining part of the proof is similar to the proof of Theorem 3, we omit the details Corollary 4 If f T P Cp, m, α, β, then pr p 1 f z pr p 1 + mp 2 1 α m pα[1 βp + βm] rm 1 mp 2 1 α m pα[1 βp + βm] rm 1 where z = r < 1 The result is sharp for fz given by 25 We shall now prove the following closure theorems for the class T S gp, m, α Theorem 5 Let λ k 0 for k = 1, 2,, l and l k=1 λ k 1 If the functions F k z defined by 26 F k z = z p f n,k z n are in the class T Sgp, m, α for every k = 1, 2,, l, then the function fz defined by l fz = z p λ k f n,k z n is in the class T S gp, m, α Proof Since F k z T S gp, m, α, it follows from Theorem 21 that k=1 27 n pαf n,k b n
for every k = 1, 2,, l Hence l n pα λ k f n,k b n = k=1 Functions with negative coefficients 185 l λ k n pαf n,k b n k=1 l k=1 By Theorem 1, it follows that fz T S gp, m, α Corollary 5 The class T S gp, m, α is closed under convex linear combinations Theorem 6 Let F p z := z p and F n z := z p p1 α n pαb n z n for n = m, m + 1, The function fz T Sgp, m, α if and only if fz can be expressed in the form 28 fz = λ p z p + λ n F n z where λ n 0 for n = p, m, m + 1, and λ p + λ n = 1 Proof If the function fz is expressed in the form given by 28, then fz = z p λ n z n and for this function, we have n pα λ nb n = p1 αλ n = p1 α1 λ p p1 α By Theorem 1, we have fz T S gp, m, α Conversely, let fz T S gp, m, α From Theorem 2, we have Therefore by taking λ k for n = m, m + 1, and λ n := n pαa nb n λ p := 1 for n = m, m + 1, λ n,
186 R M Ali, M H Khan, V Ravichandran, and K G Subramanian we see that fz is of the form given by 28 Theorem 7 Let hz = z p + h nz n with h n > 0 i Let 1 αnh n and β := inf n m [ n pαbn 1 αnh n 1 αph n If f T Sgp, m, α, then f T Sh p, m, β ii If f T Sgp, m, α, then f T Sh p, m, β in z < rα, β, where { [ ] 1 } n pα 1 β b n n p rα, β := min 1, inf n m n pβ 1 α or Proof i From the definition of β, it follows that β 1 αnh n 1 αph n n pβh n 1 β and therefore, in view of 21, n pβ p1 β a nh n 1 α h n ] n pα a nb n 1 This completes the proof of i ii It is easy to see that f satisfies 1 zf h p R z > β z < r f hz if and only if 29 n pβa n h n r n p p1 β From the definition of rα, β, we have 210 n pβ p1 β h nr n p n pα b n and the result now follows from 210, 29 and 21
Functions with negative coefficients 187 Theorem 7 contains several results For example, when p = 1, m = 2, hz = z/1 z and gz = z/1 z 2, the class T Sg1, 2, α consists of convex functions of order α in T Theorem 7i yields the order of starlikeness, ie, β = 2/3 α Similarly, when p = 1, m = 2, hz = z/1 z 2, gz = z/1 z, and β = 0, we get the radius of convexity for starlike functions of order α in T These results were proved by Silverman [4] We now prove that the class T Sgp, m, α is closed under convolution with certain functions and give an application of this result to show that the class T Sgp, m, α is closed under the familiar Bernardi integral operator Theorem 8 Let hz = z p + h nz n be analytic in with 0 h n 1 If fz T S gp, m, α, then f hz T S gp, m, α Proof The result follows by a straight forward application of Theorem 1 The generalized Bernardi integral operator is defined by 211 F z = c + p z c Since we have the following: z 0 F z = fz t c 1 ftdt z p + c > 1; z c + p c + n zn, Corollary 6 If fz T S gp, m, α, then F z given by 211 is also in T S gp, m, α References [1] O P Ahuja, Hadamard products of analytic functions defined by Ruscheweyh derivatives, in: Current topics in analytic function theory, 13 28,H M Srivastava, S Owa, editors, World Sci Publishing, Singapore, 1992 [2] V Ravichandran, On starlike functions with negative coefficients, Far East J Math Sci 8 2003, no 3, 359 364 [3] G St Sǎlǎgean, Subclasses of univalent functions, in Complex analysis: fifth Romanian-Finnish seminar, Part I Bucharest, 1981, 362 372, Lecture Notes in Mathematics 1013, Springer-Verlag, Berlin and New York, 1983 [4] H Silverman, Univalent functions with negative coefficients, Proc Amer Math Soc 51 1975, 109 116
188 R M Ali, M H Khan, V Ravichandran, and K G Subramanian Rosihan M Ali, School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia E-mail: rosihan@csusmmy M Hussain Khan, Department of Mathematics, Islamiah College, Vaniambadi 635 751, India V Ravichandran, School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia E-mail: vravi@csusmmy K G Subramanian, Department of Mathematics, Madras Christian College, Tambaram, Chennai-600 059, India E-mail: kgsmani@vsnlnet