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Trigonometric Convexity for the Multidimensional Indicator after Ivanov

Aleksandr Mkrtchyan, Armen Vagharshakyan

Abstract

Multidimensional indicator after Ivanov is a generalization of the notion of indicator, that is well-known for analytic functions in one complex variable, to analytic functions in several complex variables. We prove an analogue of trigonometric convexity for it. Additionally, we show that our estimate is sharp. The proof is based on the multidimensional analogue of the sectorial Fourier inversion formula.

Trigonometric Convexity for the Multidimensional Indicator after Ivanov

Abstract

Multidimensional indicator after Ivanov is a generalization of the notion of indicator, that is well-known for analytic functions in one complex variable, to analytic functions in several complex variables. We prove an analogue of trigonometric convexity for it. Additionally, we show that our estimate is sharp. The proof is based on the multidimensional analogue of the sectorial Fourier inversion formula.
Paper Structure (15 sections, 3 theorems, 68 equations, 5 figures)

This paper contains 15 sections, 3 theorems, 68 equations, 5 figures.

Table of Contents

  1. Introduction.
  2. Main results.
  3. Indicators of functions in one complex variable.
  4. Indicators of functions in several complex variables.
  5. Two-dimensional sectorial Fourier inversion formula.
  6. Two-dimensional concatenated Laplace transform
  7. Verification of the sectorial Fourier inversion formula.
  8. Estimates for two-dimensional indicator after Ivanov.
  9. A calculation relating to figure \ref{['fig_c']} on page \ref{['fig_c']}.
  10. An auxiliary estimate.
  11. Step 1.
  12. Step 2.
  13. Step 3.
  14. Proof of trigonometric convexity.
  15. Proof of sharpness.

Key Result

Theorem 1.1.5

Let a function $f \in Exp\left(\alpha_1,\alpha_2\right)$ and the numbers $A^+_1,A^+_2,A^-_1,A^-_2$ satisfy: Then we have where the constants $C_1,C_2$ are determined by the following formulas:

Figures (5)

  • Figure 1: The set $\Omega$
  • Figure 2: The curve $\Gamma_1$
  • Figure 3: Construction of $C_1$
  • Figure 4: Construction of $\Lambda_1$
  • Figure 5: $T_f\left(\theta_1,\theta_2\right)$

Theorems & Definitions (16)

  • Definition 1.1.1
  • Definition 1.1.2
  • Definition 1.1.3
  • Definition 1.1.4
  • Theorem 1.1.5
  • Remark 1.1.6
  • Definition 1.1.7
  • Definition 1.1.8
  • Theorem 1.1.9
  • Remark 1.1.10
  • ...and 6 more