This article contains the Hahn Banach theorem in the vector space over the field of complex numbers. provide the terminology and notation for this paper. The following propositions are true: (1) For every element z of C holds ||z|| = |z|. (2) For all elements x 1 , y 1 , x 2 , y 2 of R holds (x 1 + y 1 i) · (x 2 + y 2 i) = (x 1 · x 2 − y 1 · y 2) + (x 1 · y 2 + x 2 · y 1)i. (3) For every real n...

I referred her to [2], where Theorem 3.6 asserts that K1(R) is R /Ẽ, where Ẽ is the group generated by (1 + xy)/(1 + yx) with x, y in R and 1 + xy in R and where the last sentence in §3 says that Ẽ is not [R, R] in the case when R = M2(Z/2Z) is the ring of 2 by 2 matrices over a field of two elements (Z is the ring of integers). Moreover, in this case the group K1(R) is trivial while R/[R , R] ...

Let f (z) be an entire function and M(r) the maximum of I f (z) on z = r . We give some results on the density of the set of points at which f (z) I is small in comparison with M(r) ; although simple, these results seem not to have been noticed before . If E is a measurable set in the z-plane, we denote by DR (E) the ratio m(z c E, I z < R)/1rRZ and by D(E) and D(E) the upper and lower densitie...

The notation and terminology used in this paper are introduced in the following articles: [16], [13], [2], [3], [5], [1], [7], [9], [12], [10], [8], [18], [14], [11], [6], [15], and [17]. For simplicity, we use the following convention: x, r, a, x0, p are real numbers, n, i, m are elements of N, Z is an open subset of R, and f , f1, f2 are partial functions from R to R. Next we state a number o...

One can prove the following proposition (1) For every complex number z and for every real number r holds <(r ·z) = r · <(z) and =(r · z) = r · =(z). Let S be a finite sequence of elements of C. The functor <(S) yielding a finite sequence of elements of R is defined by: (Def. 1) <(S) = <((S qua partial function from N to C)). The functor =(S) yielding a finite sequence of elements of R is define...

Let β > 0 and let α be an integer which is at least 2. If f is an analytic function in the unit disc D which has power series representation f(z) =∑∞k=0akzkα , limsupk→∞(log+ |ak|/ logk) = α(1+β), then the first author has proved that f is unbounded in every sector {z ∈D :φ− < argz < φ+ , for > 0}. A natural conjecture concerning these functions is that limsupr→1−(logL(r)/ logM(r)) > 0, where L...