© Hindawi Publishing Corp. MAPPING PROPERTIES FOR CONVOLUTIONS INVOLVING HYPERGEOMETRIC FUNCTIONS

For μ≥0, we consider a linear operator Lμ:A→A defined by the convolution fμ∗f, 
where fμ=(1−μ)z2F1(a,b,c;z)


Introduction. Let A denote the class of functions of the form
a n z n , (1.1) which are analytic in the open unit disk U = {z : |z| < 1} and S denotes the subclass of functions in A which are univalent in U.Moreover, let S * (α) and K(α) be the subclasses of S consisting, respectively, of functions which are starlike of order α and convex of order α, where 0 ≤ α < 1 in U. Clearly, we have S * (α) ⊆ S * (0) = S * , where S * denotes the class of functions in A which are starlike in U and K(α) ⊆ K(0) = K, where K denotes the class of functions in A which are convex in U , and we mention the well-known inclusion chain K ⊂ S * (1/2) ⊂ S * ⊂ S. For the analytic functions g and h on U with g(0) = h(0), g is said to be subordinate to h if there exists an analytic function w on U such that w(0) = 0, |w(z)| < 1, and g(z) = h(w(z)) for z ∈ U. We denote this subordinated relation by For −1 ≤ A < B ≤ 1, a function p, which is analytic in U with p(0) = 1, is said to belong to the class P (A,B) if The above condition means that p takes the values in the disk with a center (1−AB)/(1−B 2 ) and a radius |A−B|/(1−B 2 ).The boundary circle cuts the real axis at the points (1+A)/(1+B) and The class ϕ * (A, B) was introduced by N. Shukla and P. Shukla [4].Also, Janowski [2] introduced the class P (A,B).For the fixed natural number n, the subclass P n (A, B) of P (A,B) containing functions p of the form p(z) = 1 + p n z n + •••, z ∈ U , was defined by Stankiewicz and Waniurski [7].In addition, Stankiewicz and Trojnar-Spelina [6] The class R η (β) was studied by Kanas and Srivastava [3].The hypergeometric function 2 F 1 (a,b,c; z) is given as a power series, converging in U , in the following way where a, b, and c are complex numbers with c = 0, −1, −2,..., and (λ) n denotes the Pochhammer symbol (or the generalized factorial since (1) n = n!) defined, in terms of the Gamma function Γ , by (1.7) Note that 2 F 1 (a,b,c; z), for a = c and b = 1 (or, alternatively, for a = 1 and b = c), reduces to the relatively more familiar geometric function.We also note that 2 F 1 (a,b,c;1) converges for Re(c − a − b) > 0 and is related to the Gamma functions by The Hadamard product (or convolution) of two power series f (z) = ∞ n=0 a n z n and g(z) = ∞ n=0 b n z n is defined as the power series (1.9) N. Shukla and P. Shukla [4] studied the mapping properties of a function f µ to be as given in and investigated the geometric properties of an integral operator of the form We now consider a linear operator L µ : A → A defined by (1.12) For µ = 0 in (1.12), L µ (f ) = [I a,b,c (f )](z), which was introduced by Hohlov [1].Also, Kanas and Srivastava [3], and Srivastava and Owa [5] showed that the operator I a,b,c (f ) is the natural extensions of the Alexander, Libera, Bernardi, and Carlson-Shaffer operators.In this paper, we find a relation between R η (β) and ϕ * (A, B) involving the operator L µ (f ).Furthermore, we study to obtain some conditions for the starlikeness and convexity of the convolution of I and f , which are given by (1.11) and (1.1), respectively, for f ∈ R η (β).

Main results.
We make use of the following lemma.
Lemma 2.1 [4].Sufficient conditions for f of the form (1.1) to be in ϕ * (A, B) and Proof.By Lemma 2.1, it suffices to show that (2.4) Now, this last expression is bounded above by B − A if (2.2) holds.
Proof.The proof follows from Lemma 2.1.Using the method of the proof of Theorem 2.2, we omit the details involved.
In our next theorems, we find the sufficient conditions for I * f to be in ϕ * (A, B) and K(A, B).From the definition of I given by (1.11), we obtain Proof.By Lemma 2.1, it satisfies to show that Suppose that f ∈ R η (β).Then by (1.5) we observe that by (2.11).This completes the proof.
Proof.The proof follows from Lemma 2.1 and by applying similar method as in the proof of Theorem 2.8; we omit the details involved.

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

Corollary 2 . 10 .
Let a > 1, b > 1, and c > a+b.If f ∈ S and the inequality

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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