Generalization of the Pecaric-Rajic Inequality in a Quasi-Banach Space

In the present paper, we shall give an extension of the well known PecaricRajic inequality in a quasi-Banach space, we establish the generalized inequality for an arbitrary number of finitely many nonzero elements of a quasi-Banach space, and obtain the corresponding upper and lower bounds. As a result, we get some more general inequalities.


Introduction
Let us first recall some basic facts concerning quasi-Banach spaces and some preliminary results.For more information about quasi-Banach spaces, the readers can refer to [1].
Definition 1 Let X be a linear space.A quasi-norm is a real-valued function on X satisfying the following: 1. 0 x ≥ for all x X ∈ and 0 x = if and only if x x for all λ ∈  and all x X ∈ ; 3. There is a constant 1 K ≥ such that ( ) The pair ( ) , X ⋅ is called a quasi-normed space if ⋅ is a quasi-norm on X .
A quasi-Banach space is a complete quasi-normed space.
A quasi-norm ⋅ is called a p-norm ( ) Let X be a normed linear space.The following is the well known Dunkl- Williams inequality (see [2]), which states that the for any two nonzero elements Many authors have studied this inequality over the years, and various refinements of this inequality (1) have been obtained (see e.g [3] [4] [5]).Pecaric and Rajic [6] got the following inequality in a normed linear space. , .
Furthermore, the authors [6] also showed that these inequalities imply some refinements of the generalized triangle inequalities obtained by some authors.
For generalized triangle inequalities, note that, some authors have also got many related results (see [7] [8]).In this paper, we shall discuss some extensions of the inequalities ( 2) and ( 3) for an arbitrary number of finitely many nonzero elements of a quasi-Banach space.

Main Results
Note that, given a p-norm, the formula ( ) gives us a translation invariant metric on X .By the Aoki-Rolewicz theorem [9] (see also [1]), each quasi-norm is equivalent to some p-norm.Henceforth we can get similar results with p-norm.In the following, we first generalize the inequalities (2) and (3) with p-norm a p-Banach space.
Theorem 2 Let X be a p-Banach space and 1 , , n a a  nonzero elements of X .Then we have Proof.First, let us prove the inequality (4): for a fixed  which is the inequality (4).The second inequality (5) follows likewise and the details are omitted.Now, we generalize the inequalities ( 2) and ( 3) with quasi-norm in a quasi-Banach space.
Theorem 3 Let X be a quasi-Banach space and 1 , , n a a  nonzero elements of X .Then we have .
where C is a constant and 1 C ≥ .
Proof.First, us prove the inequality (6): for a fixed Hence, in order to get the inequality (6), let us set , where C ≥ for all 1 j n ≤ ≤ .Thus, from the above inequality it follows that  which is the inequality (6).
In order to proof the second inequality (7), we proceed in a similar way.For a fixed { } where 1 1 C ≥ .From this it follows that Hence, in order to proof the inequality (7), let us set , where C ≥ for all 1 j n ≤ ≤ .Thus, from the above inequality it follows that 1 1 Thus, from the above inequality we can get .
This completes the proof.

Conclusion
In this paper we establish a generalisation of the so-called Pecaric-Rajic inequality by providing upper and lower bounds for the norm of the linear combination , where 1 , , n a a  nonzero elements of X .Further- more, we also obtain the corresponding inequalities in a p-Banach space with pnorm.We should also indicate that when 1 C = in Theorem 3, the inequalities (2) and (3) can be obtained as a particular case of the results established in Theorem 3. Thus, we get some more general inequalities.