We construct a generalized C r-Fractal Interpolation Function (C r-FIF) f by prescribing any combination of r values of the derivatives f (k) , k = 1, 2,. .. , r, at boundary points of the interval I = [x 0 , x N ]. Our approach to construction settles several questions of Barnsley and Harrington [J. Approx Theory, 57 (1989), pp. 14-34] when construction is not restricted to prescribing the values of f (k) at only the initial endpoint of the interval I. In general, even in the case when r equations involving f (k) (x 0) and f (k) (x N), k = 1, 2,. .. , r, are prescribed, our method of construction of the C r-FIF works equally well. In view of wide ranging applications of the classical cubic splines in several mathematical and engineering problems, the explicit construction of cubic spline FIF f Δ (x) through moments is developed. It is shown that the sequence {f Δ k (x)} converges to the defining data function Φ(x) on two classes of sequences of meshes at least as rapidly as the square of the mesh norm Δ k approaches to zero, provided that Φ (r) (x) is continuous on I for r = 2, 3, or 4.

Abstract. We derive bounds on the variance of a finite universe. Some related inequalities for the roots of the polynomial equations and bounds for the largest and smallest eigenvalues of a square matrix with real spectrum are obtained.Mathematics subject classification (2010): 60E15, 15A42.

In the present paper, the coefficient estimates are found for the class S * −1 (α) consisting of inverses of functions in the class of univalent starlike functions of order α in D = {z ∈ C: |z| < 1}. These estimates extend the work of Krzyz, Libera and Zlotkiewicz [J.G. Krzyz, R.J. Libera, E. Zlotkiewicz, Coefficients of inverse of regular starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 33 (10) (1979) 103-109] who found sharp estimates on only first two coefficients for the functions in the class S * −1 (α). The coefficient estimates are also found for the class Σ * −1 (α), consisting of inverses of functions in the class Σ * (α) of univalent starlike functions of order α in V = {z ∈ C: 1 < |z| < ∞}. The open problem of finding sharp coefficient estimates for functions in the class Σ * (α) stands completely settled in the present work by our method developed here.

We describe the dynamical behaviour of the entire transcendental non-critically finite function $f_\lambda (z) = \lambda(e^z - 1)/z$, $\lambda > 0$. Our main result is to obtain a computationally useful characterization of the Julia set of $f_\lambda (z)$ as the closure of the set of points with orbits escaping to infinity under iteration, which in turn is applied to the generation of the pictures of the Julia set of $f_\lambda (z)$. Such a characterization was hitherto known only for critically finite entire transcendental functions [11]. We find that bifurcation in the dynamics of $f_\lambda (z)$ occurs at $\lambda = \lambda^{*}$ ($\approx 0.64761$) where $\lambda^\ast = {(x^{*})}^{2} /({e}^{x^{*}} -1)$ and $x^{*}$ is the unique positive real root of the equation $e^{x}(2 -x ) -2 = 0$.

We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ4 and its projection in ℝ3 is taken. The extra degree of freedom coming from ℝ4 provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.