The theory of inhomogeneous analytic and polynomial materials is developed. These are media where the coefficients entering the equations involve analytic functions or polynomials. Three types of analytic or polynomial materials are identified. The first two types involve an integer p. If p takes its maximum value then we have a complete analytic or polynomial material. Otherwise it is incomple...

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In this paper, we introduce a method to obtain the dual sequence of a given polynomial set using the lowering operator associated with the involved polynomials. As application, we derive polynomial expansions of analytic functions. The particular case corresponding to Boas–Buck polynomials allows us to unify many polynomial expansions of analytic functions in the literature. This method can be ...

Harmonic and analytic functions have natural discrete analogues. Harmonic functions can be defined on every graph, while analytic functions (or, more precisely, holomorphic forms) can be defined on graphs embedded in orientable surfaces. Many important properties of the “true” harmonic and analytic functions can be carried over to the discrete setting. We prove that a nonzero analytic function ...

Let P (x, z) = z + ∑d i=1 ai(x)z d−i be a polynomial, where ai are real analytic functions in an open subset U of R. If for any x ∈ U the polynomial z 7→ P (x, z) has only real roots, then we can write those roots as locally lipschitz functions of x. Moreover, there exists a modification (a locally finite composition of blowing-ups with smooth centers) σ : W → U such that the roots of the corre...

In this paper, we use the Faber polynomial expansion to obtain bounds for general coefficients a n of bi-univalent functions in family analytic open unit disk. Estimation bound value initial these classes is also established.