## Lower bounds for Mahler measures of certain special polynomials. (Minorations pour les mesures de Mahler de certains polynômes particuliers.)(French)Zbl 1006.11007

Let $$P(x) = ax^d + \dots \pm b$$ be a polynomial of degree $$d \geq 1$$ with real coefficients and with $$a, b > 0$$. Let $$p = |P(1)|$$, $$q = |P(-1)|$$ and $$r = |P(i)|$$, and let $$M$$ denote the Mahler measure of $$P$$. The author proves in a very simple way a number of interesting inequalities, for example that if all of the roots of $$P$$ are positive and if $$p > 0$$ then $$M^{1/d} \geq (p^{1/d} + q^{1/d})/2$$, and if all roots are real and $$p,q,r > 0$$ then $$M^{2/d} \geq ((pq)^{1/d} + r^{2/d})/2$$. He shows that these are stronger than the related results of V. Flammang [J. Théor. Nombres Bordx. 9, 69–74 (1997; Zbl 0892.11035)]. The main ingredient in the proof is the following simple consequence of the arithmetic-geometric mean inequality: if $$a_i, b_i > 0$$ for $$i = 1,\dots,n$$ then $(a_1\cdots a_n)^{1/n} + (b_1 \cdots b_n)^{1/n} \leq ((a_1+b_1)\cdots(a_n+b_n))^{1/n}.$

### MSC:

 11C08 Polynomials in number theory 26D05 Inequalities for trigonometric functions and polynomials

Zbl 0892.11035
Full Text:

### References:

 [1] Bertin, M.J., Exposé au Cinquième Congrès de L’Association Canadienne de Théorie des Nombres. Ottawa, août 1996. [2] Flammang, V., Inégalités sur la mesure de Mahler d’un polynôme. Journal de Théorie des Nombres de Bordeaux9 (1997), 69-74. · Zbl 0892.11035 [3] Schinzel, A., On the product of the conjugates outside the unit circle of an algebraic. Acta Arith.24 (1973), 385-399. · Zbl 0275.12004
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