In this paper, we study the entire solutions of two quadratic functional equations in the complex plane. One consists of three basic terms, $ f(z), f'(z) $ and $ f(z+c) $, and the other one consists of $ f(z), f'(z) $ and $ f(qz) $. These two equations can be transformed into functional equations of Fermat-type. We prove that if these two equations admit finite order transcendental entire solutions, then the solutions of these two equations are both exponential functions, and their exponents are one degree polynomials, whose coefficients of the first degree term are closely related to the coefficients of the functional equation. Moreover, examples are given to show that the theorems are true. The feature of this paper is that the Fermat-type equations contain three quadratic terms, while the equations that have been studied in the previous articles in this field contain only two quadratic terms. The addition of $ f(qz) $ will make the proof methods in this paper very different from those in the existing literature. The proof becomes more difficult, and the number of cases that need to be discussed becomes much larger. In addition, when dealing with the analytical property of $ f $, we also use a different method from the previous literature.
Citation: Guowei Zhang. The exact transcendental entire solutions of complex equations with three quadratic terms[J]. AIMS Mathematics, 2023, 8(11): 27414-27438. doi: 10.3934/math.20231403
In this paper, we study the entire solutions of two quadratic functional equations in the complex plane. One consists of three basic terms, $ f(z), f'(z) $ and $ f(z+c) $, and the other one consists of $ f(z), f'(z) $ and $ f(qz) $. These two equations can be transformed into functional equations of Fermat-type. We prove that if these two equations admit finite order transcendental entire solutions, then the solutions of these two equations are both exponential functions, and their exponents are one degree polynomials, whose coefficients of the first degree term are closely related to the coefficients of the functional equation. Moreover, examples are given to show that the theorems are true. The feature of this paper is that the Fermat-type equations contain three quadratic terms, while the equations that have been studied in the previous articles in this field contain only two quadratic terms. The addition of $ f(qz) $ will make the proof methods in this paper very different from those in the existing literature. The proof becomes more difficult, and the number of cases that need to be discussed becomes much larger. In addition, when dealing with the analytical property of $ f $, we also use a different method from the previous literature.
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Guowei Zhang. The exact transcendental entire solutions of complex equations with three quadratic terms[J]. AIMS Mathematics, 2023, 8(11): 27414-27438. doi: 10.3934/math.20231403
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