We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural numbers whenever s > 2k + 2k − 3.

Let x, e > 0, uo < ... u O be real numbers. Let f be a real valued function and let A (h; u, w)f (x) h-d be a difference quotient associated with a generalized Riemann derivative. Set I = (x + uoh, x + Ud+eh) and let f have its ordinary (d 1)st derivative continuous on the closure of I and its dth ordinary derivative f('I) existent on 1. A necessary and sufficient condition that a ...

The point (M,f(M)) is called an absolute maximum of f if f(x) ≤ f(M) for every x in the domain of f . The point (m, f(m)) is called an absolute minimum of f if f(x) ≥ f(m) for every x in the domain of f . More than one absolute maximum or minimum may exist. For example, if f(x) = |x| for x ∈ [−1, 1] then f(x) ≤ 1 and there are absolute maxima at (1, 1) and at (−1, 1), but only one absolute mini...

The motivation for this paper comes from other papers treating the fractional derivatives. We introducea new definition of derivative which obeys classical properties including linearity, product rule,&nbsp;quotient rule, power chain Rolle’s theorem, mean value theorem and Taylor series. Usage thedefined is given in example section shows how our can be used solving&nbsp;differential equations. ...

We present a new proof of generalized Flett’s mean value theorem due to Pawlikowska (from 1999) using only the original Flett’s mean value theorem. Also, a Trahan-type condition is established in general case.