## 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:

  Bertin, M.J., Exposé au Cinquième Congrès de L’Association Canadienne de Théorie des Nombres. Ottawa, août 1996.  Flammang, V., Inégalités sur la mesure de Mahler d’un polynôme. Journal de Théorie des Nombres de Bordeaux9 (1997), 69-74. · Zbl 0892.11035  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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