On the zeros of lacunary-type polynomials
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On the zeros of lacunary-type polynomials Horst Alzer1 · Man Kam Kwong2 · Gradimir V. Milovanovi´c3,4 Received: 4 July 2020 / Accepted: 17 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let p ≥ 2 be an integer, M > 0 be a real number and C( p, M) = z n + an− p z n− p + · · · + a1 z + a0 max
0≤ j≤n− p
|a j | = M, n = p, p + 1, . . . ,
where the coefficients a j ( j = 0, 1, . . . , n − p) are complex numbers. Guggenheimer (Am Math Mon 71:54–55, 1964) and Aziz and Zargar (Proc Indian Acad Sci 106:127–132, 1996) proved that if P ∈ C( p, M), then all zeros of P lie in the disk |z| < δ( p, M), where δ( p, M) is the only positive solution of x p − x p−1 = M. We show that δ( p, M) is the best possible value. Moreover, we present some monotonicity/concavity/convexity properties and limit relations of δ( p, M). Keywords Polynomials · Zeros · Optimal bound · Monotonic · Concave · Convex
1 Introduction Finding bounds for the zeros of polynomials is a classical problem which attracted (and still attracts) the attention of numerous mathematicians. A well-known result
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Gradimir V. Milovanovi´c [email protected] Horst Alzer [email protected] Man Kam Kwong [email protected]
1
Waldbröl, Germany
2
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
3
The Serbian Academy of Sciences and Arts, Belgrade, Serbia
4
Faculty of Scienes and Mathematics, University of Niš, Niš, Serbia
123
H. Alzer et al.
published by Cauchy in 1829 states that if P(z) = z n + an−1 z n−1 + · · · + a1 z + a0 is a complex polynomial of degree n, then all zeros of P lie in the disk |z| ≤ 1 +
max |a j |.
0≤ j≤n−1
In the literature, various refinements of Cauchy’s theorem and many related results on the location of the zeros of polynomials are given. For more information on this subject we refer to Milovanovi´c et al. [4, Chapter 3] and Rahman and Schmeisser [5, Chapter 8], as well as the references cited therein. Our work has been inspired by two interesting papers of Guggenheimer [2] and Aziz and Zargar [1], who studied the following class of lacunary-type polynomials, C( p, M) = z n + an− p z n− p + · · · + a1 z + a0 max |a j | = M, n = p, p + 1, . . . . 0≤ j≤n− p
Here, p ≥ 2 is an integer, M > 0 is a real number and a j ( j = 0, 1, . . . , n − p) are complex numbers. They proved the following result. Proposition 1.1 Let P ∈ C( p, M). All zeros of P lie in disk |z| < δ( p, M),
(1.1)
where δ( p, M) is the only positive solution of x p − x p−1 = M.
(1.2)
Remark 1.2 All solutions of the algebraic equation (1.2) for p ≤ 4 can be obtained in symbolic form using the well-known software packages Mathematica, Maple or Matlab. In this paper all computations were performed in Mathematica, Ver. 12.1.1, on MacBook Pro (2017), OS Catalina Ver. 10.15.5 in both, symbolic and numerical mode. The corresponding command in Mathematica for p = 2 is TeXForm[x/.Solve[xˆ2-x-M == 0, x]] and then we obtain the two solutions in TeXForm 1 √ √ 1 1 − 4M
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