Using the results of Miller and Mocanu , Bulboaca  considered certain classes of firstorder differential superordinations as well as superordination-preserving integral operators . Ali et al. , have used the results of Bulboaca  to obtain sufficient conditions for normalized analyticfunctions to satisfy:
The purpose of the present paper is to derive some applications of firstorder differential subordination and superordination results involving Hadamard product for multivalent analyticfunctions 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  proved that f ∈ Ꮾ if and only if f ∈ BMOA, which easily implies a result of Baernstein II  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]).
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.
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;
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  (see also , , , ). Furthermore, judging by the re- markable flood of papers on the subject (see, for example, ,  and ). Not much is known about the bounds on the general coefficient |a n | . In the literature, there are
In 1963, Yu  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  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
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 analyticfunctions. 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
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 ).
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:
Following the earlier works (based upon the familiar concept of a neighborhood of analyticfunctions) 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 analyticfunctions 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 .
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.