Holomorphic Functions with Positive Real Part

1982 ◽  
Vol 34 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Eric Sawyer
Keyword(s):  
Unit Ball ◽  
Hardy Spaces ◽  
Convex Cone ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Spaces Of Holomorphic Functions ◽  
Extreme Element ◽  
Image Position ◽  
Special Case

The main purpose of this note is to prove a special case of the following conjecture.Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functionsis in Hp(B2) for 0 < p < 3/2.

scholarly journals Green's formula and the Hardy-Stein identities

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 135-153 ◽  
Author(s):  
Miroslav Pavlovic
Keyword(s):  
New York ◽  
Hardy Space ◽  
Unit Ball ◽  
Unit Disk ◽  
Hardy Spaces ◽  
London Math ◽  
Analogous Result ◽  
Holomorphic Functions ◽  
Spaces Of Holomorphic Functions ◽  
Complex Ball

This is a collection of some known and some new facts on the holomorphic and the harmonic version of the Hardy-Stein identity as well as on their extensions to the real and the complex ball. For example, we prove that if f is holomorphic on the unit disk D, then ??f ??Hp = ?f(0)?p + ?D?f'(z)? p-2 ?f'(z)?2(1-?z?) dA(z), (?) where Hp is the p-Hardy space, which improves a result of Yamashita [Proc. Amer. Math. Soc. 75 (1979), no. 1, 69-72]. An extension of (?) to the unit ball of Cn improves results of Beatrous an Burbea [Kodai Math. J. 8 (1985), 36-51], and of Stoll [J. London Math. Soc. (2) 48 (1993), no. 1, 126-136]. We also prove the analogous result for the harmonic Hardy spaces. The proofs of known results are shorter and more elementary then the existing ones, see Zhu [Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005, Ch. IV]. We correct some constants in that book and in a paper of Jevtic and Pavlovic [Publ. Inst. Math. (Beograd) (N.S.) 64(78) (1998), 36-52].

Boundary Interpolation for Continuous Holomorphic Functions

1989 ◽  
Vol 41 (5) ◽  
pp. 769-785
Author(s):  
William S. Cohn
Keyword(s):  
Unit Ball ◽  
Homogeneous Polynomial ◽  
Holomorphic Functions ◽  
Polynomial Expansion ◽  
Boundary Interpolation ◽  
Radial Derivative ◽  
Spaces Of Holomorphic Functions ◽  
Image Position

Let Bn denote the unit ball in Cn with boundary S. We will be concerned with spaces of holomorphic functions on Bn and will use much of the notation and terminology found in W. Rudin's book [16]. Thus, if f is holomorphic in Bn and has homogeneous polynomial expansionthe radial derivative of f is given by

Some Extreme Rays of the Positive Pluriharmonic Functions

1979 ◽  
Vol 31 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Ball ◽  
Extreme Point ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Compact Open Topology ◽  
Open Unit Ball ◽  
Pluriharmonic Functions ◽  
Image Position ◽  
Extreme Rays

1.1. We will denote by B the open unit ball in Cn, and we will denote by H(B) the class of all holomorphic functions on B. LetThus N(B) is convex (and compact in the compact open topology). We think that the structure of N(B) is of interest and importance. Thus we proved in [1] that if(1.1)if(1.2)and if n≧ 2, then g is an extreme point of N(B). We will denote by E(B) the class of all extreme points of N(B). If n = 1 and if (1.2) holds, then as is well known g ∈ E(B) if and only if(1.3)

scholarly journals ESSENTIAL NORM OF EXTENDED CESÀRO OPERATORS FROM ONE BERGMAN SPACE TO ANOTHER

2011 ◽  
Vol 85 (2) ◽  
pp. 307-314 ◽  
Author(s):  
ZHANGJIAN HU
Keyword(s):  
Unit Ball ◽  
Bergman Space ◽  
Holomorphic Functions ◽  
Essential Norm ◽  
Normal Weight ◽  
Cesàro Operator ◽  
Radial Derivative ◽  
Image Position ◽  
Cesaro Operator

AbstractLet Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which $\|f\|^p_{p,\varphi }= \int _B |f(z)|^p \varphi (z) \,dA(z)\lt +\infty $, where φ is a given normal weight. Let Tg be the extended Cesàro operator with holomorphic symbol g. The essential norm of Tg as an operator from Ap (φ) to Aq (φ) is denoted by $\|T_g\|_{e, A^p (\varphi )\to A^q (\varphi )} $. In this paper it is proved that, for p≤q, with 1/k=(1/p)−(1/q) , where ℜg(z) is the radial derivative of g; and for p>q, with 1/s=(1/q)−(1/p) .

Instability theorems for certain fifth-order differential equations

1978 ◽  
Vol 84 (2) ◽  
pp. 343-350 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Trivial Solution ◽  
Real Root ◽  
Order Differential Equation ◽  
Positive Real Part ◽  
Auxiliary Equation ◽  
Positive Real ◽  
Image Position ◽  
Fifth Order

1. Consider the constant-coefficient fifth-order differential equation:It is known from the general theory that the trivial solution of (1·1) is unstable if, and only if, the associated (auxiliary) equation:has at least one root with a positive real part. The existence of such a root naturally depends on (though not always all of) the coefficients a1, a2,…, a5. For example, ifit is clear from a consideration of the fact that the sum of the roots of (1·2) equals ( – a1) that at least one root of (1·2) has a positive real part for arbitrary values of a2,…, a5. A similar consideration, combined with the fact that the product of the roots of (1·2) equals ( – a5) will show that at least one root of (1·2) has a positive real part iffor arbitrary a2, a3 and a4. The condition a1 = 0 here in (1·4) is however superfluous whenfor then X(0) = a5 < 0 and X(R) > 0 if R > 0 is sufficiently large thus showing that there is a positive real root of (1·2) subject to (1·5) and for arbitrary a1, a2, a3 and a4.

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