Non-real zeros of derivatives of meromorphic functions

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane.     

The zeros of Euler's Psi function and its derivatives

We investigate the locations of the points of inflexion of Euler's Psi function, and the positions of the stationary points of its derivative. We also establish some trigonometric approximations to Psi which lead to improved estimates for the positions of its zeros. Finally we consider the behaviour of the horizontal separation between the branches.     

On the family of meromorphic functions whose derivatives omit a holomorphic function

In this paper,we continue to study the normality of a family of meromorphic functions without simple zeros and simple poles such that their derivatives omit a given holomorphic function.Such a family in general is not normal at the zeros of the omitted function.Our main result is the characterization of the non-normal sequences,and hence some known results are its corollaries.     

Distribution of zeros of certain meromorphic functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1428–1430, October, 1989.     

Estimates for the zeros of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimated accurately.     

On the derivative of meromorphic functions with multiple zeros

Let f be a transcendental meromorphic function and let R be a rational function, R?0. We show that if all zeros and poles of f are multiple, except possibly finitely many, then f′−R has infinitely many zeros. If f has finite order and R is a polynomial, then the conclusion holds without the hypothesis that poles be multiple.     

The logarithmic derivative of a meromorphic function

A well-known lemma on the logarithmic derivative for a function f(z), f(0) = 1 (0 < r="> m( r,\fracf￠f ) < ln+ { \fracT(r,f)r\fracrr- r } + 5.8501.m\left( {r,\frac{{f'}}{f}} \right)< \ln + \left\{ {\frac{{T(\rho ,f)}}{r}\frac{\rho }{{\rho - r}}} \right\} + 5.8501.     

Normality criteria of meromorphic functions with multiple zeros

Take positive integers n,k?2. Let F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k. If, for each pair (f,g) in F, fn(f(k)) and gn(g(k)) share a non-zero complex number a ignoring multiplicity, then F is normal in D.     

Normal families of meromorphic functions with multiple zeros

Let F be a family of meromorphic functions defined in a domain D such that for each f∈F, all zeros of f(z) are of multiplicity at least 3, and all zeros of f^′(z) are of multiplicity at least 2 in D. If for each f∈F, f^′(z)−1 has at most 1 zero in D, ignoring multiplicity, then F is normal in D.     

The imaginary zeros of a mixed Bessel function

In the present work we study the existence and monotonicity properties of the imaginary zeros of the mixed Bessel functionM _v(z)=(z²+)J_v(z)+zJ_v(z). Such a function includes as particular cases the functionsJ _v(z)(==0), J_v(z)(=–v²,=1)x andH _v(z)=J_v(z)+zJ_v(z), whereJ _v(z) is the Bessel function of the first kind and of orderv>–1 andJ _v(z), J_v(z) are the first two derivatives ofJ _v(z). Upper and lower bounds found for the imaginary zeros of the functionsJ _v(z), J_v(z) andH _v(z) improve previously known bounds.

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Zusammenfassung Dieser Artikel betrifft die Existenz und Monotonie von Eigenschaften imaginärer Nullen der gemischten BesselfunktionM _v(z)=(z²+)J_v(z)+zJ_v(z). Eine solche Funktion enthält als Spezialfall die FunktionenJ _v(z)(==0), J_v(z)(=–v²,=1) undH _v(z)=J_v(z)+zJ_v(z), woJ _v(z)die Besselfunktion von erster Art und Ordnungv>–1 andJ _v(z), J_v(z) sind die erste und zweite Ableitung vonJ _v(z). Untere und obere Schranken, die für die imaginären Nullen der FunktionenJ _v(z), J_v(z) undH _v(z) gefunden wurden, verbessern früher bekannte Resultate.

     

The zeros of the complementary error function

We show that the complementary error function, $\text{erfc} (z)= \frac{2}{\sqrt{\pi}}\int_z^{\infty}{e^{-s^2} \text{d}s}$ , has no zeros in $\text{D}= \left\{ z : \frac{3}{4} \ \pi \le Arg z \le\frac{5}{4} \ \pi \right\}$ .     

On the logarithmic derivative of a meromorphic function

We derive the following estimate for the quantity m(r,f′/tf) of the Nevanlinna theory of the distribution of values characterizing the growth of the logarithmic derivative of a meromorphic functionf(z),f(0) = 1, 0 < r < R < ∞: m(r,f′/f) < 1n+ [T(R,f)/r (R/R?r)²] + 6.0084. This estimate is more accurate than that obtained earlier by Vu Ngoyan and I. V. Ostrovskii.     

Normality of meromorphic functions with multiple zeros and shared values

In this paper, we study the normality of a family of meromorphic functions and general criteria for normality of families of meromorphic functions with multiple zeros concerning shared values are obtained.     

On zeros and fixed points of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+1)−f(z). A number of results are proved concerning the existences of zeros and fixed points of g(z) or g(z)/f(z) which expand results of Bergweiler and Langley [W. Bergweiler, J.K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007) 133-147].     

On the zeros of a polynomial and its derivatives

If is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of to a nearest zero of ? We obtain bounds for this distance depending on degree. We also show that this distance is equal to for polynomials of degree 3 and polynomials with real zeros.