In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginzburg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this report, we c...In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginzburg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this report, we consider the Liouville type theorems for Lichnerowicz equation and Ginzburg-Landau equations in Rn via two approaches from the use of maximum principle and the monotonicity展开更多
In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always e...In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.展开更多
The aim of this article is twofold.One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method.The other is to obtain the p...The aim of this article is twofold.One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method.The other is to obtain the precise values of Bloch constants for certain log-p-harmonic mappings.These results improve upon the corresponding results given in Bai et al.(Complex Anal.Oper.Theory,13(2):321-340,2019).展开更多
Let f(z)=a_o+a_1z+… be holomorphic in the unit disk {z: |z|<1} and omit the values 0 and 1. It is proved in this paper that |a_1|≤2|a_0|{|log|a_0||+Γ~4(1/4)/4π~2-mRe(a_n^8 + 1)} where m>0.04 is a constant, ...Let f(z)=a_o+a_1z+… be holomorphic in the unit disk {z: |z|<1} and omit the values 0 and 1. It is proved in this paper that |a_1|≤2|a_0|{|log|a_0||+Γ~4(1/4)/4π~2-mRe(a_n^8 + 1)} where m>0.04 is a constant, ε=1 as |a_0|≤1 and ε=-1 as |a_0|>1. This result is a precise version of the well-known theorem of Landau and an improvement of the results of W. Lai~[1], J. A. Hempel~[2] and J. A. Jenkins~[3]展开更多
文摘In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginzburg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this report, we consider the Liouville type theorems for Lichnerowicz equation and Ginzburg-Landau equations in Rn via two approaches from the use of maximum principle and the monotonicity
文摘In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.
基金supported by Guangdong Natural Science Foundation(2018A030313508)。
文摘The aim of this article is twofold.One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method.The other is to obtain the precise values of Bloch constants for certain log-p-harmonic mappings.These results improve upon the corresponding results given in Bai et al.(Complex Anal.Oper.Theory,13(2):321-340,2019).
基金Project supported by the National Natural Science Foundation of China
文摘Let f(z)=a_o+a_1z+… be holomorphic in the unit disk {z: |z|<1} and omit the values 0 and 1. It is proved in this paper that |a_1|≤2|a_0|{|log|a_0||+Γ~4(1/4)/4π~2-mRe(a_n^8 + 1)} where m>0.04 is a constant, ε=1 as |a_0|≤1 and ε=-1 as |a_0|>1. This result is a precise version of the well-known theorem of Landau and an improvement of the results of W. Lai~[1], J. A. Hempel~[2] and J. A. Jenkins~[3]
