共查询到20条相似文献,搜索用时 31 毫秒
1.
Zhigang Peng 《Journal of Mathematical Analysis and Applications》2008,340(1):209-218
Let B denote the set of functions ?(z) that are analytic in the unit disk D and satisfy |?(z)|?1(|z|<1). Let P denote the set of functions p(z) that are analytic in D and satisfy p(0)=1 and Rep(z)>0(|z|<1). Let T denote the set of functions f(z) that are analytic in D, normalized by f(0)=0 and f′(0)=1 and satisfy that f(z) is real if and only if z is real (|z|<1). In this article we investigate the support points of the subclasses of B, P and T of functions with fixed coefficients. 相似文献
2.
Janos Galambos 《Journal of Number Theory》1977,9(3):338-341
Let ?(N) > 0 be a function of positive integers N and such that ?(N) → 0 and N?(N) → ∞ as N → + ∞. Let NνN(n:…) be the number of positive integers n ≤ N for which the property stated in the dotted space holds. Finally, let g(n; N, ?, z) be the number of those prime divisors p of n which satisfy NZ?(N) ? p ? N?(N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, limvN(n : g(n; N, ?, z) ? g(n + 1; N, ?z) = k) exists and we determine its actual value. The case k = 0 induced the present investigation. Our solution for this value shows that the natural density of those integers n for which n and n + 1 have the same number of prime divisors in the range (1) exists and it is positive. 相似文献
3.
Machiel van Frankenhuijsen 《Journal of Number Theory》2005,115(2):360-370
If the Riemann zeta function vanishes at each point of the finite arithmetic progression {D+inp}0<|n|<N (D?1/2, p>0), then N<13p if D=1/2, and N<p1/D-1+o(1) in general. 相似文献
4.
We consider functionsf(z),z∈D, of one complex variable that satisfy the following weakened asymptotic monogeny condition: for some positiveσ<1/2,f(z) is monogenic at each pointξ∈D with respect to some setG(ξ) such that the lower density ofG(ξ) atξ is greater than 1/2+σ. We show that if for somep σ ≥1 the function (log+|?(z)|) p σ is locally integrable inD with respect to the plane Lebesgue measure, thenf(z) is holomorphic inD. 相似文献
5.
Minoru Murata 《Journal of Functional Analysis》1982,49(1):10-56
Let ?iA = ?i(p(D) + V) be a dissipative operator in L2(Rn), where p(D) is an elliptic differential operator of order m with real constant coefficients and V is a compact operator from the weighted Sobolev space Hm′,s (Rn) to H0, p + s (Rn), s?R, for some m′ < m ? 1 and p > 1. Let R(z) be the resolvent of A. Then an asymptotic expansion of R(z) as z approaches a critical value of the polynomial p(ξ) is given; the coefficient operators in the expansion are computed explicitly. By using the resolvent expansion and the results of M. Murata [9], an asymptotic expansion of e?itA as t → ∞ is given. 相似文献
6.
It is shown that, for solid caps D of heat balls in ? d + 1 with center z 0 = (0, 0), there exist Borel measurable functions w on D such that inf w(D) > 0 and ∫ v(z)w(z) dz ≤ v(z 0), for every supertemperature v on a neighborhood of D?. This disproves a conjecture by N. Suzuki and N.A. Watson. On the other hand, it turns out that there is no such volume mean density, if the bounded domain D in ? d × (?∞, 0) is only slightly wider at z 0 than a heat ball. 相似文献
7.
Let f(z) be a normalized convex (starlike) function on the unit disc D. Let , where z=(z1,z2,…,zn), z1∈D, , pi?1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z1),f′(z1)1/p2z2,…,f′(z1)1/pnzn) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f′(z1))1/pi|z1=0=1, i=2,…,n. Some other related results are proved. 相似文献
8.
The first part of this paper is devoted to the study of FN{\Phi_N} the orthogonal polynomials on the circle, with respect to a weight of type f = (1 − cos θ)
α
c where c is a sufficiently smooth function and ${\alpha > -\frac{1}{2}}${\alpha > -\frac{1}{2}}. We obtain an asymptotic expansion of the coefficients F*(p)N(1){\Phi^{*(p)}_{N}(1)} for all integer p where F*N{\Phi^*_N} is defined by
F*N (z) = zN [`(F)]N(\frac1z) (z 1 0){\Phi^*_N (z) = z^N \bar \Phi_N(\frac{1}{z})\ (z \not=0)}. These results allow us to obtain an asymptotic expansion of the associated Christofel–Darboux kernel, and to compute the
distribution of the eigenvalues of a family of random unitary matrices. The proof of the results related to the orthogonal
polynomials are essentially based on the inversion of the Toeplitz matrix associated to the symbol f. 相似文献
9.
Zongming Guo 《Journal of Differential Equations》2005,211(1):187-217
The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δpu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C1((0,∞)?{z2})∩C0[0,∞) satisfying f(0)=f(z1)=f(z2)=0 with 0<z1<z2, f<0 in (0,z1)∪(z2,∞), f>0 in (z1,z2). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2. 相似文献
10.
Let ? and f be functions in the Laguerre-Pólya class. Write ?(z)=e−αz2?1(z) and f(z)=e−βz2f1(z), where ?1 and f1 have genus 0 or 1 and α,β?0. If αβ<1/4 and ? has infinitely many zeros, then ?(D)f(z) has only simple real zeros, where D denotes differentiation. 相似文献
11.
Qian Lu 《Journal of Mathematical Analysis and Applications》2008,340(1):394-400
We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z)≠1, for each z∈Δ, such that there not only exists a positive number ε0 such that |an(a(z)−1)−1|?ε0 for arbitrary sequence of integers an(n∈N) and for any z∈Δ, but also exists a positive number B>0 such that for every f(z)∈F, B|f′(z)|?|f(z)| whenever f(z)f″(z)−a(z)(f′2(z))=0 in Δ. Then is normal in Δ. 相似文献
12.
Flávio Dickstein 《Journal of Differential Equations》2006,223(2):303-328
We study the Cauchy problem for the nonlinear heat equation ut-?u=|u|p-1u in RN. The initial data is of the form u0=λ?, where ?∈C0(RN) is fixed and λ>0. We first take 1<p<pf, where pf is the Fujita critical exponent, and ?∈C0(RN)∩L1(RN) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<ps, where ps is the Sobolev critical exponent, and ?(x) decaying as |x|-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial. 相似文献
13.
E. G. Goluzina 《Journal of Mathematical Sciences》2006,137(3):4774-4779
The paper studies the region of values Dm,1(T) of the system {ƒ(z1), ƒ(z2), …, ƒ(zm), ƒ(r)}, m e 1, where zj (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a2z2+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set Dm,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication,
the region of values of ƒ(zm) in the subclass of functions from the class T with prescribed values ƒ(zk) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005. 相似文献
14.
15.
彭志刚 《数学物理学报(B辑英文版)》1999,19(4):457-462
1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={… 相似文献
16.
A. S. Makin 《Differential Equations》2017,53(6):719-724
We obtain asymptotic formulas uniform with respect to the index p > 0 for the Hankel functions H p (j) (z) (j = 1, 2) for large |z| in the complex domain. These formulas generalize those known for the real argument. 相似文献
17.
El Maati Ouhabaz 《Semigroup Forum》1998,57(3):305-314
m
σ/|α|=|β|=m Dα(a
αβDβ) is a higher order degenerate-elliptic operator on L2(RN) and V ∈ L1(RN) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m. 相似文献
18.
Mostafa A. Nasr 《Proceedings Mathematical Sciences》1977,85(5):367-378
LetM (α) denote the class of α-convex functions, α real, that is the class of analytic functions? (z) =z + Σ n=2/∞ a n z n in the unit discD = {z: |z | < 1} which satisfies inD the condition ?′ (z) ?(z)/z ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z f'(z)}}{{f (z)}} + a \left( {1 + \frac{{z f''(z)}}{{f' (z)}}} \right)} \right\} > 0. Let W (a) $$ denote the class of meromorphic α-convex functions. α real, that is the class of analytic functions ? (z) =z ?1 + Σ n=0/∞ b n z n inD* = {z: 0 < |z | < 1} which satisfies inD* the conditionsz?′(z)/?(z) ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z\phi ' (z)}}{{\phi (z)}} + a \left( {1 + \frac{{z\phi ''(z)}}{{\phi ' (z)}}} \right)} \right\}< 0. $$ In this paper we obtain the relation betweenM (a) and W(α). The radius of α-convexity for certain classes of starlike functions is also obtained. 相似文献
19.
Leonard Gross 《Journal of Functional Analysis》2002,190(1):38-92
A Hermitian metric, g, on a complex manifold, M, together with a smooth probability measure, μ, on M determine minimal and maximal Dirichlet forms, QD and Qmax, given by Q(f)=∫M g(grad f(z), grad f(z)) dμ(z). QD is the form closure of Q on C∞c(M) and Qmax is the form closure of Q on C1b(M). The corresponding operators, AD and Amax, generate semigroups having standard hypercontractivity properties in the scale of Lp spaces, p>1, when the corresponding form, Q, satisfies a logarithmic Sobolev inequality. It was shown by the author (1999, Acta Math.182, 159-206) that the semigroup e−tAD has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of Lp. These results are extended here to Amax. When (M, g) is not complete it is necessary that the elliptic differential operator Amax degenerate on the boundary of M. A second proof of these strong hypercontractive inequalities for both AD and Amax is given, which depends on an extension of the submean value property of subharmonic functions. The Riemann surface for z1/n and the weighted Bergman spaces in the unit disc are given as examples. 相似文献
20.
F. Peherstorfer 《Constructive Approximation》1996,12(4):481-488
LetW N(z)=aNzN+... be a complex polynomial and letT n be the classical Chebyshev polynomial. In this article it is shown that the polynomials (2aN)?n+1Tn(WN), n ∈N, are minimal polynomials on all equipotential lines for {z∈C:|W N(z)|≤1 Λ ImW N(z)=0} 相似文献