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Generalised regular variation of arbitrary order

Edward Omey, Johan Segers (2010)

Banach Center Publications

Let f be a measurable, real function defined in a neighbourhood of infinity. The function f is said to be of generalised regular variation if there exist functions h ≢ 0 and g > 0 such that f(xt) - f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) eventually leads to the relation f(xt) - f(t) = h₁(x)g₁(t) + ⋯ + hₙ(x)gₙ(t) + o(gₙ(t)), each g i being of smaller order than its predecessor g i - 1 . The function f is said to be generalised regularly varying of...

Generalization of Ehrlich-Kjurkchiev Method for Multiple Roots of Algebraic Equations

Iliev, Anton (1998)

Serdica Mathematical Journal

In this paper a new method which is a generalization of the Ehrlich-Kjurkchiev method is developed. The method allows to find simultaneously all roots of the algebraic equation in the case when the roots are supposed to be multiple with known multiplicities. The offered generalization does not demand calculation of derivatives of order higher than first simultaneously keeping quaternary rate of convergence which makes this method suitable for application from practical point of view.

Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions

Shoshana Abramovich, Slavica Ivelić, Josip Pečarić (2010)

Open Mathematics

We present integral versions of some recently proved results which improve the Jensen-Steffensen and related inequalities for superquadratic functions. For superquadratic functions which are not convex we get inequalities analogous to the integral Jensen-Steffensen inequality for convex functions. Therefore, we get refinements of all the results which use only the convexity of these functions. One of the inequalities that we obtain for a superquadratic function φ is y ¯ ⩾ φ x ¯ + 1 λ β - λ α ∫ α β φ f t - x ¯ d λ t , where x ¯ = 1 λ β - λ α ∫ α β f t d λ t and y ¯ = 1 λ β - λ α ∫ α β φ f t d λ t which under...

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