Oscillation of Solutions of Linear Differential Equations

This paper is devoted to studying the growth problem, the zeros and fixed points distribution of the solutions of linear differential equations f″＋e^-zf′＋Q（z）f=F（z）,whereQ（z）≡h（z）e^cz and c∈R.     

Exponential polynomials as solutions of certain nonlinear difference equations

Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn + L(z, f ) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)2 + q(z)f (z + 1) = p(z), where p(z), q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ C, equations of the form f(z)n + q(z)e Q(z) f(z + c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.     

The transcendental meromorphic solutions of a certain type of nonlinear differential equations

In this paper, we study the differential equations of the following form w²+R(z)²(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w²+P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w²+(z−z₀)P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w²+A(z−z₀)²(w^′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab²=1, a²=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B²=C and q^′(z)=±P(z).     

On a certain type of nonlinear differential equations admitting transcendental meromorphic solutions

We study the differential equations w ²+R(z)(w ^(k))² = Q(z), where R(z),Q(z) are nonzero rational functions. We prove

1. if the differential equation w ²+R(z)(w′)² = Q(z), where R(z), Q(z) are nonzero rational functions, admits a transcendental meromorphic solution f, then Q ≡ C (constant), the multiplicities of the zeros of R(z) are no greater than 2 and f(z) = √C cos α(z), where α(z) is a primitive of $\tfrac{1} {{\sqrt {R(z)} }}$ such that √C cos α(z) is a transcendental meromorphic function.
2. if the differential equation w ² + R(z)(w ^(k))² = Q(z), where k ? 2 is an integer and R,Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C (constant), R(z) ≡ A (constant) and f(z) = √C cos (az + b), where $a^{2k} = \tfrac{1} {A}$ .

     

Normal families and shared functions of meromorphic functions

Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f ^(k) share 0, and |f(z)| ≥ M whenever f ^(k)(z) = h(z), then F is normal in D. The condition that f and f ^(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f ^(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f ^(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.     

The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc

We consider the complex differential equations of the form

Ak(z)f(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=F(z),     

Entire functions that share a polynomial with their derivatives

Let f be a nonconstant entire function, k and q be positive integers satisfying k>q, and let Q be a polynomial of degree q. This paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial Q is shared by f and f^′ CM, and if f^(k)(z)−Q(z)=0 whenever f(z)−Q(z)=0, then f≡f^′. We give two examples to show that the hypothesis k>q is necessary.     

Rank one perturbations at infinite coupling in Pontryagin spaces

In this paper we relate the operators in the operator representations of a generalized Nevanlinna function N(z) and of the function −N(z)⁻¹ under the assumption that z=∞ is the only (generalized) pole of nonpositive type. The results are applied to the Q-function for S and H and the Q-function for S and H^∞, where H is a self-adjoint operator in a Pontryagin space with a cyclic element w, H^∞ is the self-adjoint relation obtained from H and w via a rank one perturbation at infinite coupling, and S is the symmetric operator given by S=H∩H^∞.     

Spectral analysis of a Dirac operator with a meromorphic potential

We consider an operator Q(V) of Dirac type with a meromorphic potential given in terms of a function V of the form V(z)=λV₁(z)+μV₂(z), z∈C?{0}, where V₁ is a complex polynomial of 1/z, V₂ is a polynomial of z, and λ and μ are nonzero complex parameters. The operator Q(V) acts in the Hilbert space L²(R²;C⁴)=^⊕4L²(R²). The main results we prove include: (i) the (essential) self-adjointness of Q(V); (ii) the pure discreteness of the spectrum of Q(V); (iii) if V₁(z)=z−p and 4?degV₂?p+2, then kerQ(V)≠{0} and dimkerQ(V) is independent of (λ,μ) and lower order terms of ∂V₂/∂z; (iv) a trace formula for dimkerQ(V).     

On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

A criterion of normality based on a single holomorphic function

Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f ^(k)(z)| ≤ c, where c is a constant. Then F is normal on D.     

Meromorphic Functions That Share Fixed-Points

In this paper, we prove the following result: Let f(z) and g(z) be two nonconstant meromorphic(entire) functions, n ≥ 11(n ≥ 6) a positive integer. If fⁿ(z)f′(z) and gⁿ(z)g′(z) have the same fixed-points, then either f(z) = c₁e^cz2, g(z) = c₂e^− cz2, where c₁, c₂, and c are three constants satisfying 4(c₁c₂)^n + 1c² = −1, or f(z) ≡ tg(z) for a constant t such that t^n + 1 = 1.     

An improvement of Montel's criterion

Let F be a family of holomorphic functions in a domain D, and let a(z), b(z) be two holomorphic functions in D such that a(z)?b(z), and a(z)?a^′(z) or b(z)?b^′(z). In this paper, we prove that: if, for each f∈F, f(z)−a(z) and f(z)−b(z) have no common zeros, f^′(z)=a(z) whenever f(z)=a(z), and f^′(z)=b(z) whenever f(z)=b(z) in D, then F is normal in D. This result improves and generalizes the classical Montel's normality criterion, and the related results of Pang, Fang and the first author. Some examples are given to show the sharpness of our result.     

On a nontraditional method of approximation

We study the approximation of functions f(z) that are analytic in a neighborhood of zero by finite sums of the form H _n (z) = H _n (h, f, {λ _k }; z) = Σ _k=1 ⁿ λ _k h(λ _k z), where h is a fixed function that is analytic in the unit disk |z| < 1 and the numbers λ _k (which depend on h, f, and n) are calculated by a certain algorithm. An exact value of the radius of the convergence H _n (z) → f(z), n → ∞, and an order-sharp estimate for the rate of this convergence are obtained; an application to numerical analysis is given.     

Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity

This research is a continuation of a recent paper due to the first four authors. Shared value problems related to a meromorphic function f(z) and its shift f(z+c), where c∈C, are studied. It is shown, for instance, that if f(z) is of finite order and shares two values CM and one value IM with its shift f(z+c), then f is a periodic function with period c. The assumption on the order of f can be dropped if f shares two shifts in different directions, leading to a new way of characterizing elliptic functions. The research findings also include an analogue for shifts of a well-known conjecture by Brück concerning the value sharing of an entire function f with its derivative f^′.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.     

The stability of a cubic type functional equation with the fixed point alternative

In this note we investigate the generalized Hyers-Ulam-Rassias stability for the new cubic type functional equation f(x+y+2z)+f(x+y−2z)+f(2x)+f(2y)=2[f(x+y)+2f(x+z)+2f(x−z)+2f(y+z)+2f(y−z)] by using the fixed point alternative. The first systematic study of fixed point theorems in nonlinear analysis is due to G. Isac and Th.M. Rassias [Internat. J. Math. Math. Sci. 19 (1996) 219-228].     

Meromorphic functions sharing two small functions with its derivative

In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM^∗, and share b(z)IM^∗ with g(z)=a₁(z)f(z)+a₂(z)f^′(z). And a₁(z), a₂(z) and b(z) (a₂(z),b(z)?0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.     

Hua operators and Poisson transform for bounded symmetric domains

Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We consider the Poisson transform Psf(z) for a hyperfunction f on S defined by the Poisson kernel Ps(z,u)=s(h(z,z)^n/r/²|h(z,u)^n/r|), (z,u)×Ω×S, s∈C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in Mn,m(C) (n?m), we prove that an eigenvalue equation for the second order Mn,n-valued Hua operator characterizes the image.     

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).