A general Minkowski-type inequality for two variable Gini means

Czinder, Péter

doi:10.5486/pmd.2000.2299

Cited by 12 publications

(8 citation statements)

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“…Multiplying by u and writing v = 1/u ∈ ] 0, 1[ we obtain the required inequality (14). As each inequality is this proof was equivalent with the preceding one ( 14) implies (13) completing the equivalence of these two inequalities. Scrutinizing the proof we easily see that the statement concerning strict inequalities holds.…”

Section: Theorem 7 (Criterion For the Comparison) The Comparison Ineq...mentioning

confidence: 72%

“…If 0 = c + d = a + b, then equality holds in the comparison inequality (13). (17) holds differ from this only, if c > 0, d > 0 (Figure 5) and if c < 0, d < 0 (Figure 11).…”

Section: Theorem 9 (Necessary Conditions For the Comparison) If The C...mentioning

confidence: 94%

“…is satisfied (provided that abcd(a − b)(c − d) = 0, otherwise the appropriate limits should stand in (14)). If the inequality is strict in (14) then it is also strict in (13).…”

Section: Theorem 7 (Criterion For the Comparison) The Comparison Ineq...mentioning

confidence: 99%

“…The solution of the comparison problem ( 15) is known (cf. [22], [26], [13]). To formulate this solution and some further results we introduce the following notations.…”

Section: Theorem 9 (Necessary Conditions For the Comparison) If The C...mentioning

confidence: 99%

“…For the comparison of Stolarsky means, we recall Theorem 10 (see [13], Theorem 1). The comparison inequality…”

Section: Theorem 9 (Necessary Conditions For the Comparison) If The C...mentioning

confidence: 99%

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Ratio of Stolarsky means: monotonicity and comparison

Losonczi¹

2009

Publ. Math. Debrecen

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The Stolarsky mean ([29], [30]) S a,b (x, y) of the numbers x, y > 0 with parameters a, b ∈ R is defined by S a,b (x, y) = b(x a − y a ) a(x b − y b ) 1 a−b if ab(a − b)(x − y) = 0, while for ab(a − b)(x − y) = 0 the function S a,b (x, y) is extended continuously. We study monotonicity properties of the ratio R a,b (x, y, z) := S a,b (x, y) S a,b (x, z) (a, b ∈ R, 0 < x < y < z) in the parameters a, b where 0 < x < y < z and completely solve the comparison problem R a,b (x, y, z) ≤ R c,d (x, y, z) (a, b, c, d ∈ R, 0 < x < y < z) for this ratio. This generalizes, among others, the results of C. E. M. Pearce and J. Pečarić [27] and F. Qi, Sh.-X. Chen and Ch.-P. Chen [9], [15].

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Section: Theorem 7 (Criterion For the Comparison) The Comparison Ineq...mentioning

confidence: 72%

“…If 0 = c + d = a + b, then equality holds in the comparison inequality (13). (17) holds differ from this only, if c > 0, d > 0 (Figure 5) and if c < 0, d < 0 (Figure 11).…”

Section: Theorem 9 (Necessary Conditions For the Comparison) If The C...mentioning

confidence: 94%

“…is satisfied (provided that abcd(a − b)(c − d) = 0, otherwise the appropriate limits should stand in (14)). If the inequality is strict in (14) then it is also strict in (13).…”

Section: Theorem 7 (Criterion For the Comparison) The Comparison Ineq...mentioning

confidence: 99%

“…The solution of the comparison problem ( 15) is known (cf. [22], [26], [13]). To formulate this solution and some further results we introduce the following notations.…”

Section: Theorem 9 (Necessary Conditions For the Comparison) If The C...mentioning

confidence: 99%

“…For the comparison of Stolarsky means, we recall Theorem 10 (see [13], Theorem 1). The comparison inequality…”

Section: Theorem 9 (Necessary Conditions For the Comparison) If The C...mentioning

confidence: 99%

See 3 more Smart Citations