### Abstract

We solve the so-called invariance equation in the class of two-variable Stolarsky means {S _{p q}: p, q ε R}, i.e., we find necessary and sufficient conditions on the six parameters a,b,c,d,p,q such that the identity S _{p,q}(S _{a},b(x,y),S _{c,d}(x,y)) = S _{p q}(x,y) (x,y ε R _{+}), be valid. We recall that, for pq(p - q) ≠ 0 and x ≠y, the Stolarsky mean S _{p,q}is defined by S _{p q}(x, y):=(q(x ^{p}/p(x ^{q} ^{1/pq} In the proof first we approximate the Stolarsky mean and we use the computer-algebra system Maple V Release 9 to compute the Taylor expansion of the approximation up to 12th order, which enables us to describe all the cases of the equality.

Original language | English |
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Pages (from-to) | 3219-3227 |

Number of pages | 9 |

Journal | Applied Mathematics and Computation |

Volume | 216 |

Issue number | 11 |

DOIs | |

Publication status | Published - Aug 1 2010 |

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### Keywords

- Computer algebra
- Gauss composition
- Invariance equation
- Stolarsky mean

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

**Computer aided solution of the invariance equation for two-variable Stolarsky means.** / Baják, Szabolcs; Páles, Z.

Research output: Contribution to journal › Article

*Applied Mathematics and Computation*, vol. 216, no. 11, pp. 3219-3227. https://doi.org/10.1016/j.amc.2010.04.046

}

TY - JOUR

T1 - Computer aided solution of the invariance equation for two-variable Stolarsky means

AU - Baják, Szabolcs

AU - Páles, Z.

PY - 2010/8/1

Y1 - 2010/8/1

N2 - We solve the so-called invariance equation in the class of two-variable Stolarsky means {S p q: p, q ε R}, i.e., we find necessary and sufficient conditions on the six parameters a,b,c,d,p,q such that the identity S p,q(S a,b(x,y),S c,d(x,y)) = S p q(x,y) (x,y ε R +), be valid. We recall that, for pq(p - q) ≠ 0 and x ≠y, the Stolarsky mean S p,qis defined by S p q(x, y):=(q(x p/p(x q 1/pq In the proof first we approximate the Stolarsky mean and we use the computer-algebra system Maple V Release 9 to compute the Taylor expansion of the approximation up to 12th order, which enables us to describe all the cases of the equality.

AB - We solve the so-called invariance equation in the class of two-variable Stolarsky means {S p q: p, q ε R}, i.e., we find necessary and sufficient conditions on the six parameters a,b,c,d,p,q such that the identity S p,q(S a,b(x,y),S c,d(x,y)) = S p q(x,y) (x,y ε R +), be valid. We recall that, for pq(p - q) ≠ 0 and x ≠y, the Stolarsky mean S p,qis defined by S p q(x, y):=(q(x p/p(x q 1/pq In the proof first we approximate the Stolarsky mean and we use the computer-algebra system Maple V Release 9 to compute the Taylor expansion of the approximation up to 12th order, which enables us to describe all the cases of the equality.

KW - Computer algebra

KW - Gauss composition

KW - Invariance equation

KW - Stolarsky mean

UR - http://www.scopus.com/inward/record.url?scp=84856257177&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856257177&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2010.04.046

DO - 10.1016/j.amc.2010.04.046

M3 - Article

VL - 216

SP - 3219

EP - 3227

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 11

ER -