We prove some inequalities which follow from the log-convexity of the sequence of Motzkin numbers Mn and from the log-concavity of the sequence Mn n! . Mathematics subject classification (2000): 05A20, 26D99. Keywordsandphrases: Inequalities,Motzkinnumbers, log-convexity, log-concavity, integer sequences. RE F ER EN C ES [1] M. AIGNER, Motzkin numbers, European Journal of Combinatorics 19 (1998), 663–675. [2] D. CALLAN, Notes on Motzkin and Schroeder Numbers, preprint. [3] R. DONAGHEY AND L. W. SHAPIRO, Motzkin Numbers, Journal of Combinatorial Theory series A 23 (1977), 291–301. [4] T. DOSLIC, D. SVRTAN AND D. VELJAN, Secondary Structures, submitted. [5] T. MOTZKIN, Relation between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanental preponderance and for non-associative products, Bulletin of American Mathematical Society 54 (1948), 352–360. [6] R. STANLEY, Enumerative Combinatorics II, Cambridge Univ. Press, Cambridge, 1999. [7] D. VELJAN, Combinatorial and Discrete Mathematics, Algoritam, Zagreb, 2001 (in Croatian). [8] P. R. STEIN AND M. WATERMAN, On Some New Sequences Generalizing the Catalan and the Motzkin Numbers, Discr. Math. 26 (1978), 261–272. c © , Zagreb Paper MIA-05-18 Mathematical Inequalities & Applications www.ele-math.com mia@ele-math.com

For quite a long time efforts have been made to develop processes for producing iron i.e. steel without employing conventional procedures – from ore, coke, blast furnace, iron, electric arc furnace, converter to steel. The insufficient availability and the high price of the coking coals have forced many countries to research and adopt the non-coke-consuming reduction and metal manufacturing processes (non-coke metallurgy, direct reduction, direct processes). This paper represents a survey of the most relevant processes from this domain by the end of 2000, which display a constant increase in the modern process metallurgy.