On multivariate higher order Lyapunov-type inequalities
© Ji and Fan; licensee Springer. 2014
Received: 30 September 2014
Accepted: 1 December 2014
Published: 12 December 2014
In this paper, by using the best Sobolev constant method, we obtain some new Lyapunov-type inequalities for a class of even-order partial differential equations; the results of this paper are new which generalize and improve some early results in the literature.
KeywordsLyapunov-type inequality even-order partial differential equations Sobolev constant
and the constant 4 is sharp.
There have been many proofs and generalizations as well as improvements on this inequality. For example, the authors in [1–3] generalized the Lyapunov-type inequality to the partial differential equations or systems.
In , Aktaş obtained the following results.
In this paper, we generalize Theorem A and Theorem B to a more general class of even order partial differential equations. Moreover, as we shall see by the end of this paper, Theorem 1 improves Theorem A significantly.
2 Main results
where , , , and .
The main results of this paper are the following theorems.
where is the Riemann zeta function.
3 Proofs of theorems
where , .
where is the Riemann zeta function: , .
In order to prove the above propositions, we need the following lemmas.
Lemma 5 ([, Proposition 2.1])
Lemma 6 ([, Theorem 1.2 and Corollary 1.3])
The first seven values of , and
Since the proof of Proposition 4 is similar to that of Proposition 3, we give only the proof of Proposition 3 below.
This is equivalent to (14). Thus we finished the proof of Proposition 3. □
which is (10). This finishes the proof of Theorem 1. □
The proof of Theorem 2 is similar to that of Theorem 1, so we omit it for simplicity.
The first eight values of
From Table 2 we see that increases very quickly, so Theorem 1 improves Theorem A significantly even in the special case of (4).
The authors thank the anonymous referees for their valuable suggestions and comments on the original manuscript.
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