A note involving p-valently Bazilević functions

A theorem involving p-valently Bazilevic functions is considered and then its certain consequences are given.


Introduction and definitions
Let Ꮽ n (p) be the class of normalized functions of the form which are analytic and p-valent in the unit disc ᐁ = {z ∈ C : |z| < 1}.A function f ∈ Ꮽ n (p) is said to be in the class n (p,α) if it satisfies the inequality e z f (z) f (z) > α 0 ≤ α < p, p ∈ N, z ∈ ᐁ .

Main results and their consequences
We begin with the following lemma due to Jack [2].
Copyright With the aid of the above lemma, we prove the following result.
, and also let the function Ᏼ be defined by where g ∈ n (p).If Ᏼ(z) satisfies one of the following conditions: where the value of complex power in (2.4) is taken to be as its principal value.
Proof.We define the function Ω by where We see clearly that the function Ω is regular in ᐁ and Ω(0) = 0. Making use of the logarithmic differentiation of both sides of (2.5) with respect to the known complex variable z, and if we make use of equality (2.5) once again, then we find that which yields Assume that there exists a point (2.8) Applying Lemma 2.1, we can then write (2.9) (2.12)But the inequalities in (2.11) and (2.12) contradict, respectively, the inequalities in (2.2) and (2.3).Hence, we conclude that |Ω(z)| < 1 for all z ∈ ᐁ.Consequently, it follows from (2.5) that Therefore, the desired proof is completed.
This theorem has many interesting and important consequences in analytic function theory and geometric function theory.We give some of these with their corresponding geometric properties.
First, if we choose the value of the parameter w as a real number with w := δ ∈ R \ {0} in the theorem, then we obtain the following corollary.
and let the function Ᏼ be defined by (2.1).Also, if Ᏼ satisfies the following conditions: Setting w = 1 and β = 0 in the theorem, we have the following corollary.
By taking w = 1 and β = 1 in the theorem, we obtain the following corollary.

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

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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