Functions Of Finite Lower Order Research Articles

On deviations and maximum points of algebroid functions of finite lower order

We consider the influence of the number of separated maximum points and Valiron deficiency over the magnitude of Petrenko's deviation of algebroid functions of finite lower order. Presented results are the generalization of Petrenko's and Niino's results. We also give examples showing that the estimates are sharp.

On transcendental directions of entire solutions of linear differential equations

<abstract><p>This paper is devoted to studying the transcendental directions of entire solutions of $ f^{(n)}+A_{n-1}f^{(n-1)}+...+A_0f = 0 $, where $ n(\geq 2) $ is an integer and $ A_i(z)(i = 0, 1, ..., n-1) $ are entire functions of finite lower order. With some additional conditions, the set of common transcendental directions of non-trivial solutions, their derivatives and their primitives must have a definite range of measure. Moreover, we obtain the lower bound of the measure of the set defined by the common transcendental directions of Jackson difference operator of non-trivial solutions.</p></abstract>

Transcendental Singularities for a Meromorphic Function with Logarithmic Derivative of Finite Lower Order

It is shown that two key results on transcendental singularities for meromorphic functions of finite lower order have refinements which hold under the weaker hypothesis that the logarithmic derivative has finite lower order.

On deviations, small functions and strong asymptotic functions of meromorphic functions

The paper addresses two long-standing problems: of extending the second main theorem of Nevanlinna to the case of small functions, and of finding an upper limit for the number of asymptotic functions of a function of finite lower order. Upper estimates of the sum of deviations and the numbers of strong asymptotic functions and strong functional asymptotic spots of meromorphic functions of finite lower order are presented. The structure of the set of Petrenko's deviations from small functions for meromorphic functions of finite lower order is examined. An analogue of Denjoy's question for strong asymptotic small functions of meromorphic functions of finite lower order is also considered.

On limit directions of Julia sets of entire solutions of linear differential equations

This paper is devoted to studying the limit directions of Julia sets of solutions of f(n)+An−1(z)f(n−1)+⋯+A0(z)f=0, where n(≥2) is an integer and Aj(z)(j=0,1,…,n−1) are entire functions of finite lower order. With some additional conditions on coefficients, we know that every non-trivial solution f(z) of such equations has infinite lower order, and prove that the limit directions of Julia sets of f(z) must have a definite range of measure.

On deviations and strong asymptotic functions of meromorphic functions of finite lower order

Let f be a transcendental meromorphic function of finite lower order with N ( r , f ) = S ( r , f ) , and let q ν be distinct rational functions, 1 ⩽ ν ⩽ k . For 0 < γ < ∞ put B ( γ ) : = { π γ sin π γ if γ ⩽ 0.5 , π γ if γ > 0.5 . The estimate of the lower and upper logarithmic density of the set E ( γ ) = { r : ∑ 1 ⩽ ν ⩽ k log + max | z | = r | f ( z ) − q ν ( z ) | − 1 < B ( γ ) T ( r , f ) } is presented in the paper. For a transcendental meromorphic function f of finite lower order, the estimate of the lower and upper logarithmic density of the set E ˆ ( γ ) = { r : ∑ 1 ⩽ ν ⩽ k log + max | z | = r | f ( z ) − p ν ( z ) | − 1 < ( d + 2 ) B ( γ ) T ( r , f ) } is also presented, where p ν are distinct polynomials and d : = max 1 ⩽ ν ⩽ k deg ( p ν ) . Moreover, the notions of strong asymptotic rational function and strong rational asymptotic spot are defined. The results for E ( γ ) and E ˆ ( γ ) are applied to obtain upper estimates of the number of strong asymptotic rational functions and strong rational asymptotic spots of meromorphic functions.

On deviations from rational functions of entire functions of finite lower order

On deviations from rational functions of entire functions of finite lower order

Radial distributions of Julia sets of meromorphic functions

AbstractWe consider a meromorphic function of finite lower order that has ∞ as its deficient value or as its Borel exceptional value. We prove that the set of limiting directions of its Julia set must have a definite range of measure.

The Bergweiler-Eremenko theorem for finite lower order

We generalise a theorem of Bergweiler and Eremenko concerning transcendental singularities of the inverse of a meromorphic function of finite order. Our theorem extends this result to function of finite lower order.

On deviations and defects of meromorphic functions of finite lower order

We obtain estimates for the sum of deviations and sum of defects to power 1/2 in terms of the Valiron defect of the derivative at zero. In particular, the Fuchs hypothesis (1958) is verified.

Paley problem for plurisubharmonic functions of finite lower order

For plurisubharmonic functions of lower order estimates of the growth of their maximum value on the sphere of radius with centre at the origin in terms of the growth of the Nevanlinna characteristics are obtained. These estimates are best possible for . The results are new even in the case of functions of the form , where is an entire function in , .

On behavior of gap series on curves and a cos φλ-type theorem

Let f(z) be an entire function of finite lower order λ and representable by a gap power series where A ⊂ N and . In the paper we prove that if γ is an unbounded curve which meets every circle [∣,z∣, = r] once and λΔ < 1/3 then the inequality is valid. It follows from this inequality, that such a function cannot be bounded on γ if λδ < 1/3 unless it is constant.

A Quantitative Estimate on Fixed-Points of Composite Meromorphic Functions

AbstractLet ƒ(z) be a transcendental meromorphic function of finite order, g(z) a transcendental entire function of finite lower order and let α(z) be a non-constant meromorphic function with T(r, α) = S(r,g). As an extension of the main result of [7], we prove thatwhere J has a positive lower logarithmic density.

On the magnitudes of deviations and spreads of meromorphic functions of finite lower order

We investigate the influence that the number of maximum modulus points of a meromorphic function on the circle has on the magnitudes of growth and value distribution. Sharp estimates are obtained for the corresponding magnitudes.

On the zeros of certain homogeneous differential polynomials

E. Mues [10] proved in 1978 that if a ∈ C\{1} and if f is a transcendental entire function which is not of the form f(z) = exp(αz + β) where α, β ∈ C, then f(z)f ′′(z) − af ′(z)2 has at least one zero. The case a = 0 is due to W. K. Hayman [5, Theorem 5]. As shown by examples like f(z) = cos z the conclusion need not hold if a = 1, see [10] for further examples. If we allow f to be meromorphic, then we have further exceptional values for a. In fact, an easy computation shows that if a = (n + 1)/n for some n ∈ N and if f(z) = F (z)−n for an entire function F with the the property that F ′′ has no zeros, then f(z)f ′′(z)−af ′(z)2 = −nF (z)−2n−1F ′′(z) has no zeros. It seems reasonable to conjecture if f is a transcendental meromorphic function not of the form f(z) = exp(αz + β) and if a 6= 1 and a 6= (n + 1)/n, then f(z)f ′′(z) − af ′(z)2 has at least one zero. This has recently been proved by J. K. Langley [8] in the case that a = 0 and had been obtained earlier by Mues [9] in this case for functions of finite lower order.

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then (a) λ is finite; (b) for every arbitrary numberk1>1, there existsk2>1 such thatT(k1r,f)≤k2T(r,f) for allr≥r0. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then (c) every deficient values off(z) is also its asymptotic value; (d) every asymptotic value off(z) is also its deficient value; (e) λ=μ; (f) \(\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} \)

The radial growth of entire functions with density conditions

Let f be an entire function of finite lower order λ and let δ be the maximal density of its nonzero coefficients. Extending results of Kovari and Anderson, we prove that if δλ<1 then for all θ ∊[0,2π]

Further results on the deficiencies of algebroid functions

Let f(z) be an n-valued algebroid function of finite lower order μ. In this paper, we give some further results on the deficiencies of f(z). Particularly if 0 ≤ μ ≤ 1/2, the corresponding result is best possible.

On the value distribution of composite entire functions

W. Bergweiler proved in 1990 that the equation f∘g= Q has infinitely many solutions in the complex plane, provided f and g are transcendental entire functions and Q is a non-constant polynomial. We prove a related result, using standard notations from the Nevanlinna theory: Let f be a non-linear entire function of finite order of growth, let g be a transcendental entire function of finite lower order and let Q be a non-constant entire function such that T(r,Q)=S(r,g). If II denotes the canonical product formed with the zeros of f∘g Q, then T(r,φ) ≠ S(r,g). As an immediate consequencef∘g−Q has infinitely many zeros.

Sums of deficiencies of algebroid functions

Let f(z) be an n–Valued algebroid function of finite lower order. In the present paper, we give a spread relation of f(z) and some applications of the spread relation.