An inclusion formula for derivatives

Often an estimate of a derivative or its range over an interval is desired rather than the derivative itself. Error terms for numerical approximations may sometimes be rigorously estimated in terms of a number of values of the derivative and a majorant. We indicate this and propose a technique for the strict estimation of the values of the derivatives in terms of central differences of the function and its majorant. We stress strictness and accept less accuracy.     

Mittag–Leffler Functions and the Truncated $${\mathcal {}}$$-fractional Derivative

In this paper, we introduce a new type of fractional derivative, which we called truncated \({\mathcal {V}}\)-fractional derivative, for \(\alpha \)-differentiable functions, by means of the six-parameter truncated Mittag–Leffler function. One remarkable characteristic of this new derivative is that it generalizes several different fractional derivatives, recently introduced: conformable fractional derivative, alternative fractional derivative, truncated alternative fractional derivative, M-fractional derivative and truncated M-fractional derivative. This new truncated \({\mathcal {V}}\)-fractional derivative satisfies several important properties of the classical derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag–Leffler function is a generalization of Mittag–Leffler functions of one, two, three, four and five parameters, we were able to extend some of the classical results of the integer-order calculus, namely: Rolle’s theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for derivatives and as an application we calculate the truncated \({\mathcal {V}}\)-fractional derivative of the two-parameter Mittag–Leffler function. Finally, we present the \({\mathcal {V}}\)-fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the \({\mathcal {V}}\)-fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated \({\mathcal {V}}\)-fractional derivative and the truncated \({\mathcal {V}}\)-fractional integral and the fractional derivative and fractional integral in the Riemann–Liouville sense when the order parameter \(\alpha \) lies between 0 and 1 (\(0<\alpha <1\)).     

Harmonic Bergman functions as radial derivatives of Bergman functions

In the setting of the half-space of the euclidean -space, we show that every harmonic Bergman function is the radial derivative of a Bergman function with an appropriate norm bound.

     

First-order and second-order ε-directional derivatives of a marginal function in convex programming with linear inequality constraints

Hiriart-Urruty gave formulas of the first-order and second-order -directional derivatives of a marginal function for a convex programming problem with linear equality constraints, that is, the image of a function under linear mapping (Ref. 1). In this paper, we extend his results to a problem with linear inequality constraints. The formula of the first-order derivative is given with the help of a duality theorem. A lower estimate for the second-order -directional derivative is given.The author wishes to thank Professor N. Furukawa and Dr. H. Kawasaki for their helpful comments and encouragements. He is also indebted to one referee for pointing out the proof of Proposition 3.1.     

Relations between upper bounds of absolute values of functions and their higher derivatives

In this paper we consider functionsf(t), – < t < , which are n times continuously differentiable with a given convex modulus of continuity of the n-th derivative. For a certain class of periodic functions we establish a relationship between upper bounds of the absolute values of a function and its n-th derivative.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 329–338, September, 1973.     

On preponderant differentiability of typical continuous functions

In the literature, several definitions of a preponderant derivative exist. An old result of Jarník implies that a typical continuous function on has a (strong) preponderant derivative at no point. We show that a typical continuous function on has an infinite (weak) preponderant derivative at each point from a -dense subset of .

     

Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives

We study problems of optimal recovery of functions and their derivatives in the L ₂ metric on the line from information about the Fourier transform of the function in question known approximately on a finite interval or on the entire line. Exact values of optimal recovery errors and closed-form expressions for optimal recovery methods are obtained. We also prove a sharp inequality for derivatives (closely related to these recovery problems), which estimates the th derivative of a function in the L ₂-norm on the line via the L ₂-norm of the th derivative and the -norm of the Fourier transform of the function.     

Directional derivative estimates for the optimal value function of a quasidifferentiable programming problem

This paper is concerned with the optimal value function arising in the primal decomposition of a quasidifferentiable programming problem. In particular, estimates for the upper Dini directional derivative of this function are derived. They involve certain Lagrange multipliers occurring in the necessary minimum conditions to the lower level problems. This study generalizes some previously published results on this subject.     

Symmetric and quantum symmetric derivatives of Lipschitz functions

The symmetric derivative of a real valued function f at the real number x is defined to be

     

General Fractional Opial Type Inequalities

A large variety of Lp(p 0) form very general Opial type inequalities arepresented engaging different order generalized fractional derivatives of a function. These arebased on a generalization of Taylors formula for generalized fractional derivatives. In the finalresults of this work, a monotonicity property of the involved function/highest-order generalizedfractional derivative is used.     

Necessary and sufficient conditions for the existence of liapunov function of Lur'e form with\dot V \leqslant 0

This paper is concerned with the problem of absolute stability for a control system with several executive elements. Necessary and sufficient conditions are obtained for the existence of Liapunov function of Lur'e form with negative semi-definite derivative (i.e. ).     

Packing Measure Analysis of Harmonic Measure

We show that, for any Jordan domain J in R², harmonic measureis supported by a Borel set of packing dimension 1. We alsoobtain incomplete analogs to the results of Makarov, which connectthe almost everywhere behavior of the derivative near the boundaryfor the conformal mapping function from the unit disk J withthe Hausdorff measure properties of sets supporting the harmonicmeasure.     

Differentiability of Peano derivatives

Peano differentiability is a notion of higher-order Fréchet differentiability. H. W. Oliver gave sufficient conditions for the Peano derivative to be a Fréchet derivative in the case of functions of a real variable. Here we generalize this theorem to functions of several variables.

     

The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function. II. The complex plane

Let u ? ?∞be a subharmonic function in the complex plane. We establish necessary and/or sufficient conditions for the existence of a nonzero entire function f for which the modulus of the product of each of its kth derivative k = 0, 1,..., by any polynomial p is not greater than the function Ce ^u in the entire complex plane, where C is a constant depending on k and p. The results obtained significantly strengthen and develop a number of results of Lars Hörmander (1997).     

The Fourier-Walsh series of the functions absolutely continuous in the generalized restricted sense

We consider the questions of convergence in Lorentz spaces for the Fourier-Walsh series of the functions with Denjoy integrable derivative. We prove that a condition on a function f sufficient for its Fourier-Walsh series to converge in the Lorentz spaces “near” L _∞ cannot be expressed in terms of the growth of the derivative f′.     

Transcendental values of the digamma function

Let ψ(x) denote the digamma function, that is, the logarithmic derivative of Euler's Γ-function. Let q be a positive integer greater than 1 and γ denote Euler's constant. We show that all the numbers

     

Integral inequalities for the Hilbert transform applied to a nonlocal transport equation

We prove several weighted inequalities involving the Hilbert transform of a function f(x) and its derivative. One of those inequalities,

is used to show finite time blow-up for a transport equation with nonlocal velocity.