Let p(z) be a polynomial of the form
, .
We discuss a sufficient condition for the existence of zeros of p(z) in an annulus
z ∈ ℂ: 1 - c < |z| < 1 + c,
where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.
Combining difference and q-difference equations, we study the properties of meromorphic solutions of q-shift difference equations from the point of view of value distribution. We obtain lower bounds for the Nevanlinna lower order for meromorphic solutions of such equations. Our results improve and extend previous theorems by Zheng and Chen and by Liu and Qi. Some examples are also given to illustrate our results.
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