le2 { Re d(a, f)l de > (1.2) n=2ar, 01 < e 2 ON ALPHA-CLOSE-TO-CONVEX FUNCTIONS OF ORDER BETA functions. f(z) z + z a n Re (1 ) zf + (I + zf"(z)) do >

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1 Internat. J. Math. & Math. Sci. Vol. 9 No. 3 (1986) ON ALPHA-CLOSE-TO-CONVEX FUNCTIONS OF ORDER BETA M.A. NASR Faculty of Science University of Mansoura Mansoura, Egypt (Received January 9, 1985) ABSTRACT Let MB() [ > 0 and B -> O] denote the class of all functions f(z) z + z a n z n--2 analytic in the unit disc U with f (z)f(z)/z 0 and which satisfy for z=re the condition - Re (1 ) zf + (I + zf"(z)) do > f( f (z) for all e > 2 s I. In this note we show that each fmb() is close-to-star of order m when o < (. KEY WORDS AND PHRASES. Close-to-staa, functions, close-to-convex functions, a-convex functions MATHEMATICS SUBJECT CLASSIFICATION CODE. 30 C45 I. NTRODUCTI ON. A shall denote the class of all functions f(z) =z + z z n=2ar, n analytic in the unit disc U {z-lzl<1} and S shall denote the subclass of functions in A which are univalent in U. Let a_>o, _>o, and let f A with f (z)f(z)/z 0 in b, and let J (, f) (I- ) zf (z) - + f(z) (i + zf"(z)). f"(z) (I I) If for z re i( U" le2 { Re d(a, f)l de > (1.2) whenever 0-< 01 < e 2 2, then f is said to be an -close-to-convex function of order or f MB(). The class MB() was introduced by Nasr[l].

2 436 M.A. NASR It was shown [1] that MB()CS if and only if 0-< B < o. Note that M B (1) K B, is the class of close-to-convex functions of order B introduced by Reade [2] and studied by Goodman [3] for B -> and by Pommerenke [4] for 0 _< B < 1, and MB(O) R B is the class of close-to-star functions of order introduced by Goodman [3]. Moreover Mo(o) M(o) is the class of s-convex functions introduced by Mocanu [5] and M y/o ) By(o), o > O, is the class of Bazilevi functions of order y introduced by Nasr [6]. In this note we continue the investigation of o-close-to-convex functions of order B studied in [I]. 2. RESULTS In this section we show that each f MB(o) is close-to-star of order B when 0 < o. For >_ we show each f () is close-to-convex of order when M 0 < _< o and if f M(), then f M(y) when 0 < B < y -<. We assume, unless otherwise stated, that o is a real number, that 0 r and that z=re io Also that 0 < o We shall need the following result. LEMMA I" If f MB(y), then the function h given by h(z) f(z) (zf (z)/f(z)) (2.1) belongs to RB (The powers taken are the principal values). PROOF" Let f MB(). If we choose the branch of (zf (z)/f(z)) which is equal o when z 0 a simple calculation shows that the function h defined by (2.1) belongs to RB THEOREM i- If f MB(o) then f R B PROOF" Since f MB(o) it follows from Lenma that foe } fz,( Re d-- d0 Re I h(z) } zh z) do 8 (2.2) In (2.2) we choose the branches for f(z) 1/a and h(z) a for which h(z)o/f(z) (h(z)/f(z)) (2.3) with value for z=o. If we use (2.1), (2.2) and (2.3) it is easy to prove that Of102 Re {h(z)/f(z) 1/}dO TI. (2.4) In fact, since f is univalent, we can let w f(z), z z(w) f- (w) and w pe to obtain

3 ALPHA-CLOSE-TO-CONVEX FUNCTIONS OF ORDER BETA 437 h(z). 1/a f(z)/ 1/6 I w dw II d_ /P [ h{z(pei@))] i/a p o d (pe i@ -/ dp [P d [ h(z) P d [f(z)] II TT-- o l/a 1/a p dp. Hece o h(z) Re {fi)i/a I do Re { d[h(z)]l/4 dp. The result now follows frgm (2.1) and (2.4). COROLLARY 1- If f MI(), -> 1, then f K. PROOF" Let f M6(), -> 1, then S2 42 Re {(I+ zf"(z) 1) O f,--z) } zf (z_)} de e- Re f(z) Now from THEOREM I, we have Re zf (z)/f(z) do -I, and therefore do 76. el Re (1+ de > (-1) f (z) and the proof of Corollary is complete. COROLLARY 2" If f M6(), 0 6 -< y < &, thep f M PROOF- By THEOREN 1, f R 6. Suppose there exists a y 0 < 6 y -< e, such that f N 6 (y). Then there is U for which Re + 1- de 01 f r, < 13 so2 Re { [ ]I f () do 01 f-tt)--- However, for f (z.5) 2 Re + f() (2.6)

4 438 M.A. NASR Substituting (2.5) into (2.6), we obtain 0 < 2 (I- T) [B + Re f () { [ Tf (T) ]}de < B1B2 e But (1 )<0 implies Re f() tion that f R B. Thus f e MB(y). do] -B, which contradicts the assump- REMARK- For B 0 we obtain results due to Miller, Mocanu and Reade [7]. ACKNOWLEDGMENT The author would like to thank Professor M. O. Reade for reading the manuscript and giving some helpful suggestions. REFERENCES i. NASR, M. A., Alpha-Close-to-Convex Functions of Order Beta, Mathematica READE, M. 0., The Coefficients of Close-to-Convex Functions, Duke Math. Journal 23 (1950), GOODMAN, A. W., On Close-to-Convex Functions of Higher Order, Ann. Univ. Sci. Budapest Eotvos sect. Math. 2 5 (1972), POMMERENKE, C., On the Coefficients of Close-to-Convex Functions, Michigan Math Journal, 9 (1962), MOCANU, P. T., Une Properiete de Convexite Generalisee dans la Theorie de la Rpresentation Conform, Mathematica Ii (1969) NASR, M. A., Bazilevivc Functions of Higher Order, Rev. Roum. Math. Pures er A_p_p_. 28 (1983) MILLER, S. S., MOCANU, P. T., AND READE, M. A., All s-convex Functions are Univalent and Starlike, Proc. Amer. Math Sco. 37 (1973)

5 Journal of Applied Mathematics and Decision Sciences Special Issue on Intelligent Computational Methods for Financial Engineering Call for Papers As a multidisciplinary field, financial engineering is becoming increasingly important in today s economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions. However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation Authors should follow the Journal of Applied Mathematics and Decision Sciences manuscript format described at the journal site Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at according to the following timetable: Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Guest Editors Lean Yu, AcademyofMathematicsandSystemsScience, Chinese Academy of Sciences, Beijing , China; Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; yulean@amss.ac.cn Shouyang Wang, AcademyofMathematicsandSystems Science, Chinese Academy of Sciences, Beijing , China; sywang@amss.ac.cn K. K. Lai, Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; mskklai@cityu.edu.hk Hindawi Publishing Corporation