Inequalities For Entire Functions Research Articles

Erdős-Macintyre type theorem’s for multiple Dirichlet series: exceptional sets and open problems

In the paper, we formulate some open problems related to the best description of the values of the exceptional sets in Wiman's inequality for entire functions and in the Erd\H{o}s-Macintyre type theorems for entire multiple Dirichlet series. At the same time, we clarify the statement of one \v{I}.V. Ostrovskii problem on Wiman's inequality. We also prove three propositions and one theorem. On the one hand, in a rather special case, these results give the best possible description of the values of the exceptional set in the Erd\H{o}s-Macintyre-type theorem. On the second hand, they indicate the possible structure of the best possible description in the general case.

Differential inequalities for entire functions of finite degree and rational functions with prescribed poles

We establish new differential inequalities for the entire functions of finite degree with a majorant an entire function without zeros in the lower half-plane, for the entire functions with constraints on zeros and, as a consequence, for the rational functions with prescribed poles. All cases of equality in the main results are found. The estimates obtained generalize and strengthen some inequalities by Bernstein, Gardner, and Govil for entire functions of finite degree; by Smirnov, Aziz, and Shah for algebraic polynomials; and by Borwein and Erdelyi, Aziz and Shah, and the others for rational functions.

Inequalities for Entire Functions of Finite Degree and Polynomials

The extremal properties of polynomials and entire functions of finite degree not vanishing in the upper half-plane are studied. The exact inequalities obtained complement and strengthen the results by Genchev, Gardner and Govil, Turan, and Lax. Proofs are based on a univalence condition established by Dubinin. Bibliography: 15 titles.

Hermite interpolation and an inequality for entire functions of exponential type

Hermite interpolation and an inequality for entire functions of exponential type

On a refinement of the bernstein inequality for entire functions of exponential type

We give a strikingly short proof for a recent refinement of Baustein's inequality for entire functions of exponential type.

A refinement of the bernstein inequality for entire functions of exponential type

Let f be an entire function of ecponential type bounded on the real axis and let h be its Phragmen-Lindelof indicator. A refinement of the clasical Bernstein inequality, which includes both hf (-π/2) and hf (-π/2) is obtained.

Some integral identities and inequalities for entire functions and their application to the coherent state transform

Let Φ be an entire function on C n , and for any h > 0 and r > 0 define F r = ¦Φ(z)¦ r e −2π¦z¦ 2 h . Let dμ h denote h − n times Lebesgue measure on C n . ∝ ¦▽F r s 2 ¦ 2 dμ h = nπs h ∝ F r sdμ h . From this and a logarithmic Sobolev inequality we easily deduce q n q ∥F r∥ q ⩽ p n p ∥F r∥ p for all 0 < p ⩽ q < t8 where the L p norms are taken with respect to the measure dμ h above. We apply these results to the study of the spaces A p consisting of all entire functions Φ satisfying ∝ ¦Φ(z)¦ pe −2π¦z¦ 2 h dμ h < ∞ obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in A p for 1 ⩽ p < ∞. We then apply our results to the coherent state transform, extending and simplifying some previously known results.

A new inequality for entire functions

We prove that, if f( z) is an entire function and ¦f(z)¦ ⩽ (A 1 + A 2 ¦z¦ n) exp[ax 2 + by 2 + cx + dy] , then there are numbers C 1, C 2⩾ 0, depending only on n, A 1, A 2, a, b, c, and d such that ¦f′(z)¦ ⩽ (C 1 + C 2 ¦z¦ n + 1) exp(ax 2 + by 2 + cx + dy) .

An inequality for entire functions of exponential type

On deduit une inegalite pour les fonctions entieres de type exponentiel a partir d'un resultat d'Hormander

Inequalities for Entire Functions of Exponential Type

AbstractBernstein's inequality says that if f is an entire function of exponential type τ which is bounded on the real axis thenGenchev has proved that if, in addition, hf (π/2) ≤0, where hf is the indicator function of f, thenUsing a method of approximation due to Lewitan, in a form given by Hörmander, we obtain, to begin, a generalization and a refinement of Genchev's result. Also, we extend to entire functions of exponential type two results first proved for polynomials by Rahman. Finally, we generalize a theorem of Boas concerning trigonometric polynomials vanishing at the origin.

Interpolation of quasilinear operators and inequalities for entire functions in symmetric spaces

Interpolation of quasilinear operators and inequalities for entire functions in symmetric spaces

Some Inequalities for Entire Functions

Some Inequalities for Entire Functions

Some inequalities for entire functions

For any entire functions φ ( z ) \varphi (z) and ψ ( z ) \psi (z) on C with finite norm \[ { 1 π ∫ ∫ C | f ( z ) | 2 exp ⁡ ( − | z | 2 ) d x d y } 1 / 2 &gt; ∞ , {\left \{ {\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|f(z){|^2}\exp ( - |z{|^2})dx\;dy} } } \right \}^{1/2}} &gt; \infty , \] we show that the inequality \[ 2 π ∫ ∫ C | φ ( z ) ψ ( z ) | 2 exp ⁡ ( − 2 | z | 2 ) d x d y ⩽ 1 π ∫ ∫ C | φ ( z ) | 2 exp ⁡ ( − | z | 2 ) d x d y 1 π ∫ ∫ C | ψ ( z ) | 2 exp ⁡ ( − | z | 2 ) d x d y \begin {array}{*{20}{c}} {\frac {2}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z)\psi (z){|^2}\exp ( - 2|z{|^2})\;dx\;dy} } } \hfill \\ { \leqslant \frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z){|^2}\exp ( - |z{|^2})\;dx\;dy\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\psi (z){|^2}\exp ( - |z{|^2})\;dx\;dy} } } } } \hfill \\ \end {array} \] holds. This inequality is obtained as a special case of a general result. We also refer to some properties of a tensor product of spaces of entire functions.

An inequality for entire functions

An inequality for entire functions

New Inequalities for Entire Functions

New Inequalities for Entire Functions