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Holomorphic functions on the mixed norm spaces on the polydisc
Stevo Stevic 대한수학회 2008 대한수학회지 Vol.45 No.1
We generalize several integral inequalities for analytic functions on the open unit polydisc Un = {z ∈ Cn ||z|| < 1, j = 1,...,n}. It is shown that if a holomorphic function on Un belongs to the mixed norm space [수식], where wj(ㆍ), j = 1,..., n, are admissible weights, then all weighted derivations of order |k| (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ∈ [1, ∞) and when the order is equal to one. The equivalence of these conditions is given for all p, q ∈ (0, ∞) if wj (zj) = (1 - |zj|²)αj, αj > -1, j = 1,..., n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577–587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775–782. We generalize several integral inequalities for analytic functions on the open unit polydisc Un = {z ∈ Cn ||z|| < 1, j = 1,...,n}. It is shown that if a holomorphic function on Un belongs to the mixed norm space [수식], where wj(ㆍ), j = 1,..., n, are admissible weights, then all weighted derivations of order |k| (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ∈ [1, ∞) and when the order is equal to one. The equivalence of these conditions is given for all p, q ∈ (0, ∞) if wj (zj) = (1 - |zj|²)αj, αj > -1, j = 1,..., n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577–587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775–782.
On the recursive sequence xn+1
Stevo Stevi\'c 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1-2
The boundedness, global attractivity, oscillatory and asymptotic periodicity of the positive solutions of the dierence equation of the formxn+1 = +xpn-1xpn, n= 0, 1, ... is investigated, where all the coecients are nonnegative real numbers.
A generalization of a result of Choa on analytic functions with Hadamard gaps
Stevo Stevi\'c 대한수학회 2006 대한수학회지 Vol.43 No.3
In this paper we obtain a sufficient and necessary condition foran analytic function f on the unit ball B with Hadamard gaps,that is, for f(z)=sum_{k=1}^infty P_{n_k}(z) (the homogeneouspolynomial expansion of f) satisfying n_{k+1}/n_kgeqlambda>1 for all kin N, to belong to the weightedBergman space{mathcal A}^p_a(B)=big{f;|; int_{B}|f(z)|^p (1-|z|^2)^{a}dV(z)<infty,; fin H(B)big}.We find a growth estimate for the integral mean left(int_{ptB}|f(rzeta)|^pdsigma(zeta)right)^{1/p}, and an estimate forthe point evaluations in this class of functions. Similar resultson the mixed norm space H_{p,q,a}(B) and weighted Bergman spaceon polydisc {mathcal A}^p_{veca}(U^n) are also given.
ON THE RATIONAL(${\kappa}+1,\;{\kappa}+1$)-TYPE DIFFERENCE EQUATION
Stevic, Stevo 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.24 No.1
In this paper we investigate the boundedness character of the positive solutions of the rational difference equation of the form $$x_{n+1}=\frac{a_0+{{\sum}^k_{j=1}}a_jx_{n-j+1}}{b_0+{{\sum}^k_{j=1}}b_jx_{n-j+1}},\;\;n=0,\;1,...$$ where $k{\in}N,\;and\;a_j,b_j,\;j=0,\;1,...,\;k $, are nonnegative numbers such that $b_0+{{\sum}^k_{j=1}}b_jx_{n-j+1}>0$ for every $n{\in}N{\cup}\{0\}$. In passing we confirm several conjectures recently posed in the paper: E. Camouzis, G. Ladas and E. P. Quinn, On third order rational difference equations(part 6), J. Differ. Equations Appl. 11(8)(2005), 759-777.
HOLOMORPHIC FUNCTIONS ON THE MIXED NORM SPACES ON THE POLYDISC
Stevic, Stevo Korean Mathematical Society 2008 대한수학회지 Vol.45 No.1
We generalize several integral inequalities for analytic functions on the open unit polydisc $U^n={\{}z{\in}C^n||zj|<1,\;j=1,...,n{\}}$. It is shown that if a holomorphic function on $U^n$ belongs to the mixed norm space $A_{\vec{\omega}}^{p,q}(U^n)$, where ${\omega}_j(\cdot)$,j=1,...,n, are admissible weights, then all weighted derivations of order $|k|$ (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ${\in}\;[1,\;{\infty})$ and when the order is equal to one. The equivalence of these conditions is given for all p, q ${\in}\;(0,\;{\infty})$ if ${\omega}_j(z_j)=(1-|z_j|^2)^{{\alpha}j},\;{\alpha}_j>-1$, j=1,...,n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577-587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775-782.
ON THE RECURSIVE SEQUENCE $x_{n+l} =\alpha+\frac{x_{n-1}^{p}}{x_{n}^{p}}$
STEVIC, STEVO 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.18 No.1
The boundedness, global attractivity, oscillatory and asymptotic periodicity of the positive solutions of the difference equation of the form $x_{n+l} =\alpha+\frac{x_{n-1}^{p}}{x_{n}^{p}},\;\; n = 0, 1, ...$ is investigated, where all the coefficients are nonnegative real numbers.
A GENERALIZATION OF A RESULT OF CHOA ON ANALYTIC FUNCTIONS WITH HADAMARD GAPS
Stevic Stevo Korean Mathematical Society 2006 대한수학회지 Vol.43 No.3
In this paper we obtain a sufficient and necessary condition for an analytic function f on the unit ball B with Hadamard gaps, that is, for $f(z)\;=\;{\sum}^{\infty}_{k=1}\;P_{nk}(z)$ (the homogeneous polynomial expansion of f) satisfying $n_{k+1}/n_{k}{\ge}{\lambda}>1$ for all $k\;{\in}\;N$, to belong to the weighted Bergman space $$A^p_{\alpha}(B)\;=\;\{f{\mid}{\int}_{B}{\mid}f(z){\mid}^{p}(1-{\mid}z{\mid}^2)^{\alpha}dV(z) < {\infty},\;f{\in}H(B)\}$$. We find a growth estimate for the integral mean $$\({\int}_{{\partial}B}{\mid}f(r{\zeta}){\mid}^pd{\sigma}({\zeta})\)^{1/p}$$, and an estimate for the point evaluations in this class of functions. Similar results on the mixed norm space $H_{p,q,{\alpha}$(B) and weighted Bergman space on polydisc $A^p_{^{\to}_{\alpha}}(U^n)$ are also given.