This paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $\alpha$, respectively, we solve entire solutions of the Tumura–Clunie type differential equation $f^{n}+P(z,\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $n\geq 2$ is an integer, $P(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a nontrivial solution such that $\lambda (f)<\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\not =0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l> s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$. In particular, we precisely characterize all solutions such that $\lambda (f)<\infty$ when $l=2$ and $l=4$.

It is known that, in the unit disc as well as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc ,A_{k-2}$ of

$$ \begin{align*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geqslant 2, \end{align*} $$

$D(0,R)$

$0<R\leqslant \infty $

A classical theorem of Frei states that if $A_p$ is the last transcendental function in the sequence $A_0,\ldots ,A_{n-1}$ of entire functions, then each solution base of the differential equation $f^{(n)}+A_{n-1}f^{(n-1)}+\cdots +A_{1}f'+A_{0}f=0$ contains at least $n-p$ entire functions of infinite order. Here, the transcendental coefficient $A_p$ dominates the growth of the polynomial coefficients $A_{p+1},\ldots ,A_{n-1}$ . By expressing the dominance of $A_p$ in different ways and allowing the coefficients $A_{p+1},\ldots ,A_{n-1}$ to be transcendental, we show that the conclusion of Frei’s theorem still holds along with an additional estimation on the asymptotic lower bound for the growth of solutions. At times, these new refined results give a larger number of linearly independent solutions of infinite order than the original theorem of Frei. For such solutions, we show that $0$ is the only possible finite deficient value. Previously, this property has been known to hold for so-called admissible solutions and is commonly cited as Wittich’s theorem. Analogous results are discussed for linear differential equations in the unit disc, as well as for complex difference and complex q-difference equations.

This research deals with properties of polynomial regular functions, which were introduced in a recent study concerning Wiman-Valiron theory in the unit disc. The relation of polynomial regular functions to a number of function classes is investigated. Of particular interest is the connection to the growth class Gα, which is closely associated with the theory of linear differential equations with analytic coefficients in the unit disc. If the coefficients are polynomial regular functions, then it turns out that a finite set of real numbers containing all possible maximum modulus orders of solutions can be found. This is in contrast to what is known about the case when the coefficients belong to Gα.

Sufficient conditions for the analytic coefficients of the linear differential equation

are found such that all solutions belong to a given -space, or to the Dirichlet type subspace Dp of the classical Hardy space Hp, where 0 < p ≤ 2. For 0 < q < ∞, the space consists of those functions f, analytic in the unit disc D, for which |f(z)|(1 – |z|2)q is uniformly bounded in D, and f ∈ Dp if the integral ∫D |f′(z)|p(1 – |z|2)p–1dσz converges.

We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above “special function approach” can be described by a classical Heine problem for differential equations that admit polynomial solutions.

An explicit asymptotic formula is obtained for the norms of the spectral projections of the non-self-adjoint harmonic oscillator $H$. It is deduced that the spectral expansion of $\rme^{-Ht}$ is norm convergent if and only if $t$ is greater than a certain explicit positive constant.

We prove two sharp inequalities for the growth of solutions of certain linear differential equations in the unit disk. For the proofs of these inequalities, we use the method of successive approximations and sharp estimates for the logarithmic derivatives of finite order meromorphic functions in the unit disk. These techniques can also be used to give an alternate proof of a well-known result in the plane. The sharp logarithmic derivative estimates are a corollary of general estimates, and all these estimates have independent interest.

Consider n-th order linear differential equations with meromorphic periodic coefficients of the form w(n) + Rn-1(ez)w(n-1) + … + R1(ez)w′ + R0(ez)w = 0, n ≥ 2, where Rv(t) (0 ≤ ν ≤ n – 1) are rational functions of t. Under certain assumptions, we prove oscillation theorems concerning meromorphic solutions, which contain necessary conditions for the existence of a meromorphic solution with finite exponent of convergence of the zero-sequence. We also discuss meromorphic or entire solutions whose zero-sequences have an infinite exponent of convergence, and give a new zero-density estimate for such solutions.