Let Ω be a Cartan domain of rank r and genus p and Bν , ν > p−1, the Berezin transform on Ω; the number Bνf(z) can be interpreted as a certain invariant-mean-value of a function f around z. We show that a Lebesgue integrable function satisfying f = Bνf = Bν+1f = · · · = Bν+rf , ν ≥ p, must be M-harmonic. In a sense, this result is reminiscent of Delsarte’s two-radius mean-value theorem for ordi...

We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree k for moments of order 2s, establishing strongly diagonal behaviour for 1 6 s 6 1 2 k(k + 1) − 1 3 k + o(k). In particular, as k → ∞, we confirm the main conjecture in Vinogradov’s mean value theorem for 100% of the critical interval 1 6 s 6 1 2 k(k + 1).

We develop a substantial enhancement of the efficient congruencing method to estimate Vinogradov’s integral of degree k for moments of order 2s, thereby obtaining for the first time near-optimal estimates for s > 5 8k . There are numerous applications. In particular, when k is large, the anticipated asymptotic formula in Waring’s problem is established for sums of s kth powers of natural number...

 (b− a)M, for all x ∈ [a, b] . The constant 14 is best possible in the sense that it cannot be replaced by a smaller constant. In [2], the author has proved the following Ostrowski type inequality. Theorem 2. Let f : [a, b] → R be continuous on [a, b] with a > 0 and differentiable on (a, b) . Let p ∈ R\ {0} and assume that Kp (f ) := sup u∈(a,b) { u |f ′ (u)| } < ∞. Then we have the inequality...

Some Ostrowski type inequalities via Cauchy’s mean value theorem and applications for certain particular instances of functions are given.

Let W(k, 2) denote the least number s for which the system of equations ~ _ i x ~ = ~ S = l y i ( 1 <~j~k) has a solution with ~S=lx~+l v e ~ = l y ~ +1. We show that for large k one has W(k, 2) ~< 89 k + logtog k + O(1)), and moreover that when K is large, one has W(k, 2) ~< 89 + 1) + 1 for at least one value k in the interval [K, K 4/3 +~]. We show also that the least s for which the expected...

We apply the efficient congruencing method to estimate Vinogradov’s integral for moments of order 2s, with 1 6 s 6 k − 1. Thereby, we show that quasi-diagonal behaviour holds when s = o(k), we obtain near-optimal estimates for 1 6 s 6 1 4k 2 + k, and optimal estimates for s > k − 1. In this way we come half way to proving the main conjecture in two different directions. There are consequences f...

In this paper, the generalized Taylor’s expansion is presented for fuzzy-valued functions. To achieve this aim, fuzzyfractional mean value theorem for integral, and some properties of Caputo generalized Hukuhara derivative are necessarythat we prove them in details. In application, the fractional Euler’s method is derived for solving fuzzy fractionaldifferential equations in the sense of Caputo...