Research Article Applications of Differential Subordination for Argument Estimates of Multivalent Analytic Functions

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1 Abstract and Applied Analysis Volume 04, Article ID 63470, 4 pages search Article Applications of Differential Subordination for Argument Estimates of Multivalent Analytic Functions Meng-Ting Lu, Ting Jia, Xing-Qian Ling, and Jin-Lin Liu Department of Mathematics, Yangzhou University, Yangzhou 500, China Correspondence should be addressed to Jin-Lin Liu; jlliu@yzu.edu.cn ceived November 03; Accepted 0 January 04; Published 6 February 04 Academic Editor: David Kalaj Copyright 04 Meng-Ting Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using the method of differential subordinations, we derive some properties of multivalent analytic functions. All results presented here are sharp. ThispaperisdedicatedtoProfessorMiodragMateljević on the occasion of his 65th birthday. Introduction Let A(p) denote the class of functions f of the form f = + a p+n +n (p N = {,, 3,...}), () n= which are analytic in the open unit disk D={z C: z <}. Let f and g be analytic in D. Then,we say that f is subordinate to g in D, writtenasf g, ifthere exists an analytic function w in D, such that w z and f = g(w) (z D). Ifg is univalent in D, the subordination f g is equivalent to f(0) = g(0) and f(d) g(d).letp = + p z+ be analytic in D. Then, for B<A,itisclearthat if and only if p +Az +Bz (z D) () AB p B < A B B ( <B<A ;z D), (3) p > A (B = ; z D). (4) cently, a number of results for argument properties of analytic functions have been obtained by several authors (see, e.g., [ 5]).Theobjectiveofthepresentpaperisto derive some further interesting properties of multivalent analytic functions. The basic tool used here is the method of differential subordinations. To derive our results, we need the following lemmas. Lemma (see [6, Theorem, page 776]). Let h be analytic and starlike univalent in D with h(0) = 0.Ifg is analytic in D and zg h, z g g(0) + 0 h (t) dt. (5) t Lemma (see [5, Theorem,page84]). Let 0<α, 0<α, β=(α α )/(α +α ),andc=e βπi.alsolet λ 0 a 0, λ(b+) 0, (b+) μ 0, b+ α +α, a b If q is analytic in D with q(0) = and λ 0 (q ) a +λ(q) b+ +μ(q) b+ +zq (q ) b h (z D), max {α,α }. (6) (7)

2 Abstract and Applied Analysis h =λ 0 ( +cz z ) a((α+α)/) +( +cz z ) (/)(b+)(α+α) (μ+λ( +cz z ) (α+α)/ + α +α ( z z + cz +cz )) is (close-to-convex) univalent in D, (8) π α < arg (q ) < π α (z D). (9) The bounds α and α in (9) are sharp for the function q defined by q =( +cz (α+α)/ z ). (0) mark 3 (see [5, Lemma, page 83]). The function q defined by (0)isanalyticandunivalentconvexinD and q (D) ={w:w C, π α < arg w< π α }. (). Main sults Our first result is contained in the following. Theorem 4. Let α (0, /] and β (0, ). Iff A(p) satisfies f =0(0< z <)and f (zf p) <δ (z D), () f δ isthesmallestpositiverootoftheequation α sin ( πβ )x x+( α) sin ( πβ )=0, (3) arg (f α) < π β (z D). (4) The bound β is sharp for each α (0, /]. Proof. Let g (x) =αsin ( πβ )x x+( α) sin ( πβ ). (5) We can see easily that (3)hastwopositiveroots.Sinceg(0) > 0 and g() < 0,wehave 0< α δ δ<. (6) α Put f =α+( α) p. (7) Then, from the assumption of the theorem, we can see that p is analytic in D with p(0) = and α + ( α)p =0 for all z D. Taking the logarithmic differentiations in both sides of (7), we get zf f p= ( α) zp α+( α) p, (8) ( α) zp f (zf p)= f (α + ( α) p ) (9) for all z D.Thus,inequality()isequivalentto ( α) zp δz. (0) (α + ( α) p ) By using Lemma,(0)leadsto z 0 ( α) p (t) dt δz () (α + ( α) p (t)) or to δz. () α+( α) p According to (6), ()canbewrittenas +(α/ ( α)) δz p. (3) δz Now, by taking A = (α/( α))δ and B= δin () and(3), we have arg (f α) = arg p (4) δ < arcsin ( α+αδ )=π β for all z Dbecause of g(δ) = 0.Thisproves(4). Next, we consider the function f defined by f = δz for all z D.Itiseasytoseethat f (zf f for all z D.Since f α=( α) it follows from (3)that (5) p) = δz <δ (6) +(α/ ( α)) δz, (7) δz sup arg (f z U α) = arcsin ( δ α+αδ ) = π β. (8) Hence, we conclude that the bound β is the best possible for each α (0, /]. Next, we derive the following.

3 Abstract and Applied Analysis 3 Theorem 5. If f A(p) satisfies f =0(0< z <) and { zp f (zf p)} < (z D), (9) f The bound in (3) is sharp. Proof. Let 0<< log, (30) zp > log (z D). (3) f f p =. (3) zp Then, from the assumption of the theorem we can see that p is analytic in D with p(0) = and p =0for all z D. According to (3) and(9), we have immediately that is, Now, by using Lemma,weobtain zp p +z z ; (33) z( p ) z z. (34) log ( z). (35) p Since the function log( z) is convex univalent in D and ( log ( z)) > log (z D), (36) from (35), we get inequality (3). To show that the bound in (3) cannot be increased, we consider f = log ( z) (z D). (37) It is easy to verify that the function f satisfies inequality (9). On the other hand, we have zp log (38) f as z. Now, the proof of the theorem is complete. Finally, we discuss the following theorem. Theorem 6. Let α, (0, ). Iff A(p) satisfies f =0(0< z <)and f arg { ( ( zf f p)+( ) f ) for all z D, ( )α } <πδ (39) δ= + (( α) ( ) + ) π tan ( ), (40) α ( ) f >α (z D). (4) The bound δ in (39) is sharp. Proof. Define the function p by (7). For α, (0, ), it follows from (7)and(8)that ( α) ( ( zf f p)+( ) f ) = ( )α } ( α)( ) p + for all z D.Putting λ= a=b=λ 0 =0, α =α =, α ( ) p +zp (4) ( α)( ) α ( ), μ = in Lemma and using (4), we see that if ( α) ( ( zf f p)+( ) f ) h ( )α } h, =( +z ( ) )(( α) z (4)holdstrue. + z z ), ( +z ( ) )+α z (43) (44) (45)

4 4 Abstract and Applied Analysis Letting0 <θ<πand x=cot(θ/),we deduce that arg h(e iθ ) = π ( ) + arg {( α) xe πi/ α ( ) + + i (x + x )} = π + tan ( ( ( α)( ) + ) x + ). 4α ( ) x Making use of (46), we obtain that inf z =(z =±) arg h = min 0<θ<π arg h(eiθ ) = π + min x>0 tan ( ( ( α)( ) + ) x + ) 4α ( ) x (46) ferences [] A. Gangadharan, V. Ravichandran, and T. N. Shanmugam, Radii of convexity and strong starlikeness for some classes of analytic functions, Mathematical Analysis and Applications,vol.,no.,pp.30 33,997. [] J.-L. Liu, The Noor integral and strongly starlike functions, Mathematical Analysis and Applications,vol.6,no.,pp ,00. [3] M. Nunokawa, S. Owa, E. Y. Duman, and M. Aydoǧan, Some properties of analytic functions relating to the Miller and Mocanu result, Computers & Mathematics with Applications, vol.6,no.5,pp.9 95,0. [4] N.-E. Xu, D.-G. Yang, and S. Owa, On strongly starlike multivalent functions of order β and type α, Mathematische Nachrichten,vol.83,no.8,pp.07 8,00. [5] D.-G. Yang and J.-L. Liu, Argument inequalities for certain analytic functions, Mathematical and Computer Modelling,vol. 5, no. 9-0, pp. 8 8, 00. [6] T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Mathematical Journal, vol. 37, no. 4, pp , 970. = π (( α) + ( ) + ) tan ( ) α ( ) =πδ. (47) Therefore, if f A(p) satisfies (39), the subordination (44)holds,and,thus,weobtain(4). For the function we find that f = +( α) z, (48) z ( α) ( ( zf f p)+( ) f ) ( )α }=h, (49) h is defined by (45). In view of (46) and(49), we conclude that the bound δ in (39) is the largest number such that (4)holdstrue.Thiscompletestheproof. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment The authors would like to express their sincere thanks to the referees for careful reading and suggestions which helped them improve the paper.

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