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ON A CERTAIN SUBCLASS OF STARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS Subclass of Starlike Functions A. Y. LASHIN Department of Mathematics Faculty of Science Mansoura University Mansoura, 35516, EGYPT. A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page EMail: aylashin@yahoo.com Received: 11 December, 2007 Accepted: 29 May, 2009 Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C45. Key words: Univalent functions, Starlike functions, Integral means, Neighborhoods, Partial sums. Abstract: We introduce the class H(α, β) of analytic functions with negative coefficients. In this paper we give some properties of functions in the class H(α, β) and we obtain coefficient estimates, neighborhood and integral means inequalities for the function f (z) belonging to the class H(α, β). We also establish some results concerning the partial sums for the function f (z) belonging to the class H(α, β). Acknowledgements: The author would like to thank the referee of the paper for his helpful suggestions. Contents JJ II J I Page 1 of 18 Go Back Full Screen Close Contents 1 Introduction 3 2 Coefficient Estimates 5 3 Some Properties of the Class H(α, β) 7 4 Neighborhood Results 8 5 Integral Means Inequalities 10 6 Partial Sums 12 Subclass of Starlike Functions A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page Contents JJ II J I Page 2 of 18 Go Back Full Screen Close 1. Introduction Let A denote the class of functions of the form ∞ X (1.1) f (z) = z + ak z k , k=2 which are analytic in the unit disc U = {z : |z| < 1}. And let S denote the subclass of A consisting of univalent functions f (z) in U. A function f (z) in S is said to be starlike of order α if and only if 0 zf (z) Re >α (z ∈ U ), f (z) Subclass of Starlike Functions A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page for some α (0 ≤ α < 1). We denote by S ∗ (α) the class of all functions in S which are starlike of order α. It is well-known that S ∗ (α) ⊆ S ∗ (0) ≡ S ∗ . Further, a function f (z) in S is said to be convex of order α in U if and only if 00 zf (z) Re 1 + 0 >α (z ∈ U ), f (z) for some α (0 ≤ α < 1). We denote by K(α) the class of all functions in S which are convex of order α. The classes S ∗ (α), and K(α) were first introduced by Robertson [8], and later were studied by Schild [10], MacGregor [4], and Pinchuk [7]. Let T denote the subclass of S whose elements can be expressed in the form: (1.2) f (z) = z − ∞ X k=2 ak z k (ak ≥ 0). Contents JJ II J I Page 3 of 18 Go Back Full Screen Close We denote by T ∗ (α) and C(α), respectively, the classes obtained by taking the intersections of S ∗ (α) and K(α) with T, T ∗ (α) = S ∗ (α) ∩ T and C(α) = K(α) ∩ T. The classes T ∗ (α) and C(α) were introduced by Silverman [11]. Let H(α, β) denote the class of functions f (z) ∈ A which satisfy the condition 2 00 αz f (z) zf 0 (z) Re + >β f (z) f (z) for some α ≥ 0, 0 ≤ β < 1, f (z) 6= 0 and z ∈ U. z The classes H(α, β) and H(α, 0) were introduced and studied by Obraddovic and Joshi [5], Padmanabhan [6], Li and Owa [2], Xu and Yang [14], Singh and Gupta [13], and others. Further, we denote by H(α, β) the class obtained by taking intersections of the class H(α, β) with T , that is H(α, β) = H(α, β) ∩ T. We note that H(0, β) = T ∗ (β) (Silverman [11]). Subclass of Starlike Functions A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page Contents JJ II J I Page 4 of 18 Go Back Full Screen Close 2. Coefficient Estimates Theorem 2.1. A function f (z) ∈ T is in the class H(α, β) if and only if (2.1) ∞ X [(k − 1)(αk + 1) + (1 − β)]ak ≤ 1 − β. k=2 Subclass of Starlike Functions The result is sharp. Proof. Assume that the inequality (2.1) holds and let |z| < 1. Then we have 2 00 P∞ αz f (z) zf 0 (z) − k=2 (k − 1)(αk + 1)ak z k−1 P f (z) + f (z) − 1 = k−1 1− ∞ a z k k=2 P∞ (k − 1)(αk + 1)ak P ≤ 1 − β. ≤ k=2 1− ∞ k=2 ak 00 αz 2 f (z) f (z) zf 0 (z) f (z) This shows that the values of + lie in the circle centered at w = 1 whose radius is 1 − β. Hence f (z) is in the class H(α, β). To prove the converse, assume that f (z) defined by (1.2) is in the class H(α, β). Then 2 00 αz f (z) zf 0 (z) (2.2) Re + f (z) f (z) P 1− ∞ − 1) + k)]ak z k−1 k=2 [αk(k P = Re >β k−1 1− ∞ k=2 ak z 00 for z ∈ U. Choose values of z on the real axis so that αz 2 f (z) f (z) + zf 0 (z) f (z) is real. Upon A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page Contents JJ II J I Page 5 of 18 Go Back Full Screen Close clearing the denominator in (2.2) and letting z → 1− through real values, we have ! ∞ ∞ X X [αk(k − 1) + k]ak , β 1− ak ≤ 1 − k=2 k=2 which obviously is the required result (2.1). Finally, we note that the assertion (2.1) of Theorem 2.1 is sharp, with the extremal function being Subclass of Starlike Functions A. Y. Lashin (2.3) 1−β f (z) = z − zk [(k − 1)(αk + 1) + (1 − β)] (k ≥ 2). vol. 10, iss. 2, art. 40, 2009 Title Page Corollary 2.2. Let f (z) ∈ T be in the class H(α, β). Then we have (2.4) ak ≤ 1−β [(k − 1)(αk + 1) + (1 − β)] (k ≥ 2). Equality in (2.4) holds true for the function f (z) given by (2.3). Contents JJ II J I Page 6 of 18 Go Back Full Screen Close 3. Some Properties of the Class H(α, β) Theorem 3.1. Let 0 ≤ α1 < α2 and 0 ≤ β < 1. Then H(α2 , β) ⊂ H(α1 , β). Proof. It follows from Theorem 2.1. That ∞ X [(k − 1)(α1 k + 1) + (1 − β)]ak Subclass of Starlike Functions k=2 < ∞ X A. Y. Lashin [(k − 1)(α2 k + 1) + (1 − β)]ak ≤ 1 − β vol. 10, iss. 2, art. 40, 2009 k=2 for f (z) ∈ H(α2 , β). Hence f (z) ∈ H(α1 , β). Title Page Corollary 3.2. H(α, β) ⊆ T ∗ (β). Contents The proof is now immediate because α ≥ 0. JJ II J I Page 7 of 18 Go Back Full Screen Close 4. Neighborhood Results Following the earlier investigations of Goodman [1] and Ruscheweyh [9], we define the δ− neighborhood of function f (z) ∈ T by: ) ( ∞ ∞ X X k |ak − bk | ≤ δ . bk z k , Nδ (f ) = g ∈ T : g(z) = z − k=2 k=2 Subclass of Starlike Functions In particular, for the identity function A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 e(z) = z, we immediately have Title Page ( (4.1) Nδ (e) = g ∈ T : g(z) = z − ∞ X bk z k , k=2 Theorem 4.1. H(α, β) ⊆ Nδ (e), where δ = ∞ X ) k |bk | ≤ δ Contents . k=2 2(1−β) . (2α+2−β) JJ II J I Page 8 of 18 Proof. Let f (z) ∈ H(α, β). Then, in view of Theorem 2.1, since [(k − 1)(αk + 1) + (1 − β)] is an increasing function of k (k ≥ 2), we have (2α + 2 − β) ∞ X ak ≤ k=2 ∞ X Full Screen [(k − 1)(αk + 1) + (1 − β)]ak ≤ 1 − β, k=2 which immediately yields (4.2) ∞ X k=2 Go Back ak ≤ 1−β . (2α + 2 − β) Close On the other hand, we also find from (2.1) (α + 1) ∞ X k=2 kak − β ∞ X ak ≤ k=2 = (4.3) ∞ X k=2 ∞ X [(α(k − 1) + 1)k − β)]ak [(k − 1)(αk + 1) + (1 − β)]ak ≤ 1 − β. Subclass of Starlike Functions k=2 From (4.3) and (4.2), we have (α + 1) A. Y. Lashin ∞ X kak ≤ (1 − β) + β k=2 ∞ X vol. 10, iss. 2, art. 40, 2009 ak k=2 1−β ≤ (1 − β) + β (2α + 2 − β) 2(α + 1)(1 − β) ≤ , (2α + 2 − β) that is, (4.4) Title Page Contents JJ II J I Page 9 of 18 ∞ X k=2 kak ≤ 2 (1 − β) = δ, (2α + 2 − β) which in view of the definition (4.1), prove Theorem 4.1. Letting α = 0, in the above theorem, we have: Corollary 4.2. T ∗ (β) ⊆ Nδ (e), where δ = 2(1−β) . (2−β) Go Back Full Screen Close 5. Integral Means Inequalities We need the following lemma. Lemma 5.1 ([3]). If f and g are analytic in U with f ≺ g, then Z 2π Z 2π g(reiθ )δ dθ ≤ f (reiθ )δ dθ, 0 0 Subclass of Starlike Functions where δ > 0, z = reiθ and 0 < r < 1. A. Y. Lashin Applying Lemma 5.1, and (2.1), we prove the following theorem. Theorem 5.2. Let δ > 0. If f (z) ∈ H(α, β), then for z = reiθ , 0 < r < 1, we have Z 2π Z 2π f (reiθ )δ dθ ≤ f2 (reiθ )δ dθ, 0 0 where (1 − β) z2. f2 (z) = z − (2α + 2 − β) (5.1) Proof. Let f (z) defined by (1.2) and f2 (z) be given by (5.1). We must show that δ δ Z 2π Z 2π ∞ X (1 − β) k−1 1 − dθ. ak z dθ ≤ z 1 − (2α + 2 − β) 0 0 k=2 By Lemma 5.1, it suffices to show that 1− ∞ X k=2 ak z k−1 ≺ 1 − (1 − β) z. (2α + 2 − β) vol. 10, iss. 2, art. 40, 2009 Title Page Contents JJ II J I Page 10 of 18 Go Back Full Screen Close Setting (5.2) 1− ∞ X ak z k−1 = 1 − k=2 (1 − β) w(z). (2α + 2 − β) From (5.2) and (2.1), we obtain ∞ X (2α + 2 − β) |w(z)| = ak z k−1 (1 − β) k=2 ≤ |z| Subclass of Starlike Functions A. Y. Lashin ∞ X [(k − 1)(αk + 1) + (1 − β)] 1−β k=2 vol. 10, iss. 2, art. 40, 2009 ak ≤ |z| . Title Page This completes the proof of the theorem. Contents Letting α = 0 in the above theorem, we have: ∗ iθ Corollary 5.3. Let δ > 0. If f (z) ∈ T (β), then for z = re , 0 < r < 1, we have Z 2π Z 2π δ iθ f (re ) dθ ≤ f2 (reiθ )δ dθ, 0 0 where f2 (z) = z − (1 − β) 2 z . (2 − β) JJ II J I Page 11 of 18 Go Back Full Screen Close 6. Partial Sums In this section we will examine the ratio of a function of the (1.2) to its sequence Pform n of partial sums defined by f1 (z) = z and fn (z) = z − k=2 ak z k when the coefficients of f are sufficiently the determine small to satisfy condition 0 (2.1). Wewill f (z) fn (z) f (z) fn0 (z) sharp lower bounds for Re fn (z) , Re f (z) , Re f 0 (z) and Re f 0 (z) . n In what follows, we will use the well known result that Re 1 − w(z) > 0, 1 + w(z) if and only if w(z) = ∞ X Subclass of Starlike Functions A. Y. Lashin z ∈ U, vol. 10, iss. 2, art. 40, 2009 Title Page ck z k Contents k=1 satisfies the inequality |w(z)| ≤ |z| . JJ II Theorem 6.1. If f (z) ∈ H(α, β), then J I (6.1) f (z) 1 Re ≥1− fn (z) cn+1 Page 12 of 18 (z ∈ U, n ∈ N ) and Full Screen (6.2) where ck =: Go Back Re fn (z) f (z) ≥ [(k−1)(αk+1)+(1−β)] 1−β cn+1 1 + cn+1 (z ∈ U, n ∈ N ) , . The estimates in (6.1) and (6.2) are sharp. Close Proof. We employ the same P technique used by Silverman [12]. The function f (z) ∈ H(α, β), if and only if ∞ k=2 ck ak ≤ 1. It is easy to verify that ck+1 > ck > 1. Thus, (6.3) n X ak + cn+1 ∞ X ak ≤ k=n+1 k=2 ∞ X ck ak ≤ 1. k=2 We may write P P k−1 1 − nk=2 ak z k−1 − cn+1 ∞ f (z) 1 k=n+1 ak z P cn+1 − 1− = fn (z) cn+1 1 − nk=2 ak z k−1 1 + D(z) = . 1 + E(z) Then A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page Set so that Subclass of Starlike Functions 1 + D(z) 1 − w(z) = , 1 + E(z) 1 + w(z) E(z) − D(z) w(z) = . 2 + D(z) + E(z) P k−1 cn+1 ∞ k=n+1 ak z Pn P w(z) = k−1 2 − 2 k=2 ak z k−1 − cn+1 ∞ k=n+1 ak z and P∞ |w(z)| ≤ cn+1 k=n+1 ak Pn P . 2 − 2 k=2 ak − cn+1 ∞ k=n+1 ak Now |w(z)| ≤ 1 if and only if n X k=2 ak + cn+1 ∞ X k=n+1 ak ≤ 1, Contents JJ II J I Page 13 of 18 Go Back Full Screen Close which is true by (6.3). This readily yields the assertion (6.1) of Theorem 6.1. To see that f (z) = z − (6.4) z n+1 cn+1 gives sharp results, we observe that zn f (z) =1− . fn (z) cn+1 Subclass of Starlike Functions A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Letting z → 1− , we have f (z) 1 =1− , fn (z) cn+1 which shows that the bounds in (6.1) are the best possible for each n ∈ N. Similarly, we take P P k−1 1 − nk=2 ak z k−1 + cn+1 ∞ fn (z) cn+1 k=n+1 ak z P (1 + cn+1 ) − = k−1 f (z) 1 + cn+1 1− ∞ k=2 ak z 1 − w(z) := , 1 + w(z) where P (1 + cn+1 ) ∞ k=n+1 ak Pn P |w(z)| ≤ . 2 − 2 k=2 ak + (1 − cn+1 ) ∞ k=n+1 ak Now |w(z)| ≤ 1 if and only if n X k=2 ak + cn+1 ∞ X k=n+1 ak ≤ 1, Title Page Contents JJ II J I Page 14 of 18 Go Back Full Screen Close which is true by (6.3). This immediately leads to the assertion (6.2) of Theorem 6.1. The estimate in (6.2) is sharp with the extremal function f (z) given by (6.4). This completes the proof of Theorem 6.1. Letting α = 0 in the above theorem, we have: Corollary 6.2. If f (z) ∈ T ∗ (β), then n f (z) Re ≥ , fn (z) (n + 1 − β) (z ∈ U ) Subclass of Starlike Functions A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 and n+1−β fn (z) ≥ , (z ∈ U ) . f (z) (n + 2 − 2β) The result is sharp for every n, with the extremal function Re f (z) = z − (6.5) 1−β z n+1 . (n + 1 − β) We now turn to the ratios involving derivatives. The proof of Theorem 6.3 below follows the pattern of that in Theorem 6.1, and so the details may be omitted. Theorem 6.3. If f (z) ∈ H(α, β), then Re (6.6) f 0 (z) n+1 ≥1− 0 fn (z) cn+1 (6.7) Re fn0 (z) f 0 (z) ≥ cn+1 n + 1 + cn+1 Contents JJ II J I Page 15 of 18 Go Back (z ∈ U ) , Full Screen Close and Title Page (z ∈ U, n ∈ N ) . The estimates in (6.6) and (6.7) are sharp with the extremal function given by (6.4). Letting α = 0 in the above theorem, we have: Corollary 6.4. If f (z) ∈ T ∗ (β), then Re and f 0 (z) βn ≥ , 0 fn (z) (n + 1 − β) f 0 (z) n+1−β Re n0 ≥ , f (z) n + (1 − β)(n + 2) (z ∈ U ) , (z ∈ U ) . The result is sharp for every n, with the extremal function given by (6.5). Subclass of Starlike Functions A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page Contents JJ II J I Page 16 of 18 Go Back Full Screen Close References [1] A.W. GOODMAN, Univalent functions and analytic curves, Proc. Amer. Math. Soc., 8(3) (1957), 598–601. [2] J.L. LI AND S. OWA, Sufficient conditions for starlikeness, Indian J. Pure Appl. Math., 33 (2002), 313-318. [3] J.E. LITTLEWOOD, On inequalities in the theory of functions, Proc. London Math. Soc., 23 (1925), 481–519. [4] T.H. MacGREGOR, The radius for starlike functions of order 12 , Proc. Amer. Math. Soc., 14 (1963), 71–76. Subclass of Starlike Functions A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page [5] M. OBRADOVIC AND S.B. JOSHI, On certain classes of strongly starlike functions, Taiwanese J. Math., 2(3) (1998), 297–302. [6] K.S. PADMANABHAN, On sufficient conditions for starlikeness, Indian J. Pure Appl. Math., 32(4) (2001), 543–550. [7] B. PINCHUK, On starlike and convex functions of order a, Duke Math. J., 35 (1968), 89–103. [8] M.S. ROBERTSON, On the theory of univalent functions, Ann. of Math., 37 (1936), 374–408. [9] S. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81(4) (1981), 521–527. [10] A. SCHILD, On starlike functions of order a, Amer. J. Math., 87 (1965), 65–70. [11] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109–116. Contents JJ II J I Page 17 of 18 Go Back Full Screen Close [12] H. SILVERMAN, Partial sums of starlike and convex functions, J. Math. Anal. Appl., 209 (1997), 221–227. [13] S. SINGH AND S. GUPTA, First order differential subordinations and starlikeness of analytic maps in the unit disc, Kyungpook Math. J., 45 (2005), 395–404. [14] N. XU AND D. YANG, Some criteria for starlikeness and strongly starlikeness, Bull. Korean Math. Soc., 42(3) (2005), 579–590. Subclass of Starlike Functions A. Y. Lashin vol. 10, iss. 2, art. 40, 2009 Title Page Contents JJ II J I Page 18 of 18 Go Back Full Screen Close