ON A NEW GENERALIZATION OF ALZER ’ S INEQUALITY

Let {an}n=1 be an increasing sequence of positive real numbers. Under certain conditions of this sequence we use themathematical induction and the Cauchymean-value theorem to prove the following inequality: an an+m ≤ ( (1/n) ∑n i=1ai (1/(n+m))∑n+m i=1 ai )1/r , where n and m are natural numbers and r is a positive number. The lower bound is best possible. This inequality generalizes the Alzer’s inequality (1993) in a new direction. It is shown that the above inequality holds for a large class of positive, increasing and logarithmically concave sequences.


ON A NEW GENERALIZATION OF ALZER'S INEQUALITY FENG QI and LOKENATH DEBNATH
(Received 2 April 1999 and in revised form 10 December 1999) Abstract.Let {a n } ∞ n=1 be an increasing sequence of positive real numbers.Under certain conditions of this sequence we use the mathematical induction and the Cauchy mean-value theorem to prove the following inequality: , where n and m are natural numbers and r is a positive number.The lower bound is best possible.This inequality generalizes the Alzer's inequality (1993) in a new direction.It is shown that the above inequality holds for a large class of positive, increasing and logarithmically concave sequences.

Introduction.
Several authors including Alzer [1], Sandor [8], and Ume [10] proved the following inequality: where r > 0 and n ∈ N. The proof of this inequality involves the principle of the mathematical induction and other analytical methods.
The main purpose of this paper is to further generalize inequalities (1.1) and (1.3).

Main Results
Theorem 2.1.Let n and m be natural numbers.Suppose {a 1 ,a 2 ,...} is a positive and increasing sequence satisfying for any given positive real number r and k ∈ N, then we have the inequality The lower bound of (2.2) is best possible.
Proof.The inequality (2.2) is equivalent to This is also equivalent to Then, for any given positive real number r , we have the inequality (2.2).The lower bound of (2.2) is best possible. (2.12) Further, we define Direct calculation yields (2.15) (2.17) Therefore, using the inequality (2.10) and standard arguments gives (2.18) Applying the Cauchy's mean-value theorem to the left side of inequality (2.1), it turns out that there exists one point ζ ∈ (n, n + 1) such that in which the logarithmic convexity of the sequence {a n } ∞ n=1 is used.Thus, the inequality (2.1) is proved.Corollary 2.3 [4].Let n and m be natural numbers and k a nonnegative integer.Then where r is any given positive real number.The lower bound is best possible.

Note.
Recently, some inequalities related to Alzer's inequality and the sum of powers of positive integers or sequences have been proved.For details, see Qi [6,5,3], Sándor [9], and Qi and Luo [7].

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: Let n and m be natural numbers.Suppose a = {a 1 ,a 2 ,...} is a positive and increasing sequence satisfying Hence, the lower bound of (2.2) is best possible.The proof is complete.Corollary 2.2.