Using the results of Miller and Mocanu [15], Bulboaca [4] considered certain classes of **first** **order** differential superordinations as well as superordination-preserving integral operators [5]. Ali et al. [1], have used the results of Bulboaca [4] to obtain sufficient conditions for normalized **analytic** **functions** to satisfy:

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analytic functions by means of systems of first order partial differential equations goes back at least to a paper of Picard in 1891 [I]... The solutions of that system.[r]

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The purpose of the present paper is to derive some applications of **first** **order** differential subordination and superordination results involving Hadamard product for multivalent **analytic** **functions** with linear operator defined in the open unit disk. These results are applied to obtain sandwich results.

1. Introduction. For **analytic** univalent function f in the unit disk ∆ , Pommerenke [8] proved that f ∈ Ꮾ if and only if f ∈ BMOA, which easily implies a result of Baernstein II [4] about univalent Bloch **functions**: if g(z) ≠ 0 is an **analytic** univalent function in ∆ , then log g ∈ BMOA. We know that Pommerenke’s result mentioned above was generalized to Q p spaces for all p, 0 < p < ∞ , by Aulaskari et al. (cf. [2, Theorem 6.1]).

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In this work, we extend the higher-**order** FEM [19, 23] to improve the accuracy for the computation of inhomogeneous axisymmetric cavity problems. Although the higher-**order** FEM has been applied to 3-D problems, the new contribution of this work is that it develops and applies this method to BOR cavities. Both node- and edge-based elements are used to discretize the azimuthal and meridian components of the field. Numerical results are given to demonstrate the validity and efficiency of the higher-**order** FEM.

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Subordination of two **functions** f and g is denoted by f ≺ g and defined as f(z) = g(w(z)), where w(z) is schwarz function in f . Let S, S ∗ and C denotes the subclasses of χ of univalent **functions**, starlike **functions** and convex **functions** respectively. For 0 ≤ δ < 1, S ∗ (δ) and C(δ) are the subclasses of S of **functions** f satisfies;

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Key words and phrases : Analytic functions , Multivalent functions , Coefficient estimate, Distortion theorem, Starlike functions , Convex functions, Close-to-close functions, [r]

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A function f ∈ A is said to be bi-univalent in ∆ if both f and f −1 are univalent in ∆. Let Σ denote the class of bi- univalent **functions** in ∆ given by (1.1). For a brief his- tory and interesting examples in the class Σ, see [25] (see also [5], [6], [13], [16]). Furthermore, judging by the re- markable flood of papers on the subject (see, for example, [11], [23] and [24]). Not much is known about the bounds on the general coefficient |a n | . In the literature, there are

In 1963, Yu [26] ﬁrst proved the Valiron–Knopp–Bohr formula of the associated abscis- sas of bounded convergence, absolute convergence, and uniform convergence of Laplace– Stieltjes transform. Moreover, Yu [26] also estimated the growth of the maximal molecule M u (σ , F), the maximal term μ(σ, F), by introducing the concepts of the **order** of F(s), and

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some geometric characterizations; such as univalent, starlike and bounded turning are studied. Our main tool is based on the Jack Lemma. It has proven that if (A) : A(U ) → A(U ). For future work, one can use the new operator to define new classes of **analytic** **functions**. Furthermore, for further investigations, one can study the subordination and superordination idea by employing the above integral. Additionally, it can be studied the connection between closed ideals of a Banach algebra together with closed invariant subspaces of the operator DA[f ].

In calculus and engineering mathematics courses, finding f ( n ) ( c ) ( the n -th **order** derivative value of function f ( x ) at x c ), in general, necessary goes through two procedures: Evaluating f ( n ) ( x ) ( the n -th **order** derivative of

The object of the present paper is to derive certain sufficient conditions for starlikeness of reciprocal order of analytic functions in the open unit disk.. 2010 Mathematics Subject Cla[r]

Analytic Function in Tube, Strong Boundedness, Tempered Distribos isibutional Boundary Value.. AMSMOS SUBJECT CLASSIFICATION 1970 CODES.[r]

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In case ϕ has a fixed point, but is not the identity or an elliptic disk automorphism, one can use Schwarz’s lemma in classical complex analysis to show that { ϕ [ n ] } tends to that fi[r]

Shahrood and the second author by a grant of Young Researchers and Elite Club, Malard Branch, Islamic Azad University.. References.[r]

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For two **functions** f and g **analytic** in U , we say that the function f is subordinate to g, written f ≺ g or f (z) ≺ g(z)(z ∈ U), if there exists a Schwarz function w **analytic** in U with w(0) = 0 and |w(z)| < 1(z ∈ U ) such that f(z) = g(w(z)),(z ∈ U ). In particular, if the function g is univalent in U , then f ≺ g if and only if f (0) = g(0) and f (U ) ⊂ g(U ).

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By means of Lemma . and (.), we know that (.) holds true. Combining (.) and (.), we readily get the coeﬃcient estimates asserted by Theorem .. Following the earlier works (based upon the familiar concept of neighborhood of ana- lytic **functions**) by Goodman [] and Ruscheweyh [], and (more recently) by Altintaş et al. [–], Cˇataş [], Cho et al. [], Liu and Srivastava [–], Frasin [], Keerthi et al. [], Srivastava et al. [] and Wang et al. []. Assuming that γ is given by (.) and the condition (.) of Lemma . holds true, we here introduce the δ-neighborhood of a function f ∈ of the form (.) by means of the following deﬁnition:

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Following the earlier works (based upon the familiar concept of a neighborhood of **analytic** **functions**) by Goodman [] and Ruscheweyh [], and (more recently) by Altintaş et al. [–], Cˇataş [], Frasin [], Keerthi et al. [] and Srivastava et al. [], we begin by introducing here the δ-neighborhood of a function f ∈ A m of the form (.) by means of

Let H(p(z), zp (z)) ≺ h(z) be a ﬁrst-**order** diﬀerential subordination. Then a univa- lent function q(z) is called its dominant if p(z) ≺ q(z) for all **analytic** **functions** p(z) that satisfy the diﬀerential subordination. A dominant ¯ q(z) is called the best dominant if ¯ q(z) ≺ q(z) for all dominants q(z). For the general theory of ﬁrst-**order** diﬀerential subordination and its applications, we refer to [3].

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Some papers treat the existence of solutions to systems of **first**-**order** equations on time scales. Existence results are obtained in 6, 7 under hypothesis diﬀerent from ours. However, some particular cases obtained in 7 are corollaries of our existence result for problem 1.1. Also, our existence results treat the case where the right members in 1.1 and 1.2 are Δ- Carath´eodory **functions** which are more general than continuous **functions** used for systems studied in 6, 7. Let us mention that existence of extremal solutions for infinite systems of **first**-**order** equations of time scale with Δ-Carath´eodory **functions** is established in 8.

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