Let <¡>(z2) be an even entire function of temperate exponential type, L a selfadjoint realization of -A + c(x), where A is the Laplace-Beltrami operator on a Riemannian manifold, and <¡>(L) the operator given by spectral theory. A PaleyWiener theorem on the support of ( L) is proved, and is used to show that Lu = \u on a suitable domain implies (L)u = <p(\)u, as well as a generalization o...

≤ n− 1 n . Equality only occurs for p(z) = a1z + anz n with arbitrary a1, an ∈ C \ {0}. Here a proof will be given by a variational method which recently has been used in a similar way to prove Sendov’s conjecture (s. [2]). Former results and the backgrounds of both conjectures can be found in the survey article of Schmeisser [4]. Let n > 1 be fixed and define Fn as the class of nth degree moni...

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...

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 ...

Consider an experiment in which 10 subjects receive a placebo, and another 10 receive an experimental diuretic. After 8 h, the average urine output in the placebo group is 769 mL, versus 814 mL in the diuretic group—a difference of 45 mL (Figure 1). How do we know if that difference means the drug works and is not just a result of chance? of 45 mL in the average urine output between groups unde...

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.

This paper compares the expressive power of rst-order monadic logic of order, a fundamental formalism in mathematical logic and the theory of computation, with that of two formalisms for the speciication of real-time systems, the propositional versions of duration and mean value calculi. Our results show that the propositional mean value calculus is expressively complete for monadic rst-order l...

in terms of power means and other related means have precipitated the search for similar bounds for the more general 2F1(α, β; γ; r). In an early paper, B. C. Carlson considered the approximation of the hypergeometric mean values ( 2F1(−a, b; b + c; r)) in terms of means of order t, given by Mt(s, r) := {(1 − s) + s(1 − r)t}1/t. In this note, a refinement of one such approximation is establishe...