Bulletin of Mthemticl Anlysis nd Applictions ISSN: 1821-1291, URL: http://www.bmth.org Volume 4 Issue 1 (2012), Pges 226-230. FUNCTIONS OF α-slow INCREASE (COMMUNICATED BY HÜSEYIN BOR) YILUN SHANG Abstrct. The min im of this pper is to generlize the functions of slow increse to α-slow increse for ny α > 0. We investigte some bsic properties of functions of α-slow increse. In ddition, the reltionship between functions of α-slow increse nd those of slow vrition re chrcterized. 1. Introduction Functions of α-slow increse re defined s follows. Definition 1.1. Let be function defined on the intervl [, ) such tht > 0, nd with continuous derivtive f (x) > 0. For α > 0, the function is of α-slow increse if the following condition holds: f (x) 0. (1.1) x α Note tht the specil cse of 1-slow increse is introduced recently by R. Jkimczuk [3, 4] s tool to investigte the symptotic formul of Bell numbers. Further development on the subject cn be found in e.g. [1, 5]. Typicl exmples for functions of α-slow increse re s follows: x is of α-slow increse with α < 1. ln x nd ln ln x re of α-slow increse with α 1. In the next section, we will study some bsic properties for functions of α-slow increse. 2. Some Properties Theorem 2.1. Suppose tht 0 < α 1 < α 2. If is function of α 2 -slow increse, then it is of α 1 -slow increse. Proof. It is strightforwrd to see this by using (1.1). Theorem 2.2. Let α 1, α 2, β > 0 nd C R. If nd g(x) re functions of α 1 -slow nd α 2 -slow increse, respectively, then the following sttements re true. 2000 Mthemtics Subject Clssifiction. 26A06, 26A12. Key words nd phrses. α-slow increse; slowly vrying; function. c 2012 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted June 23, 2011. Published Februry 23, 2012. 226
FUNCTIONS OF α-slow INCREASE 227 + C, C nd β re functions of α 1 -slow increse. f(x β ) is function of ((α 1 1)β + 1)-slow increse, if (α 1 1)β > 1. g(x) nd + g(x) re functions of min{α 1, α 2 }-slow increse. Proof. We prove the second sttement s n exmple. Others cn be proved similrly. By Definition 1.1, we hve x (α 1 1)β+1 d dx f(xβ ) x f(x β ) βx α1β f (x β ) x f(x β ) βy α 1 f (y) 0, y f(y) (2.1) which yields tht f(x β ) is of ((α 1 1)β + 1)-slow increse if (α 1 1)β > 1. Theorem 2.3. If is function of α-slow increse for α 1, then the following its hold. ln (i) x ln x 0; (ii) x 0 for ny β > 0; x β (iii) x f (x) 0. Proof. To show (i), we obtin by L Hôspitl s rule tht ln x ln x f (x)x f (x)x α 0, (2.2) since α 1 by our ssumption. To see (ii), let 0 < γ < β. By virtue of (2.2), we hve f (x)x/ < γ for x lrge enough. Hence, ( ) x γ f (x)x γ γx γ 1 x 2γ < 0, (2.3) for lrge x. Thus, there exists some 0 < M < such tht 0 < /x γ < M. We obtin x x β 1 x x γ 0. (2.4) xβ γ (iii) is n immedite consequence of (ii) nd (1.1). Theorem 2.4. Let C R. If is function of α-slow increse for α 1, then f(x + C) 1. (2.5) Proof. We only prove the cse C > 0, nd the cse C < 0 cn be proved likewise. Applying the Lgrnge men vlue theorem, we hve f(x + C) Cf (ξ), (2.6) for some x < ξ < x + C. Combining (2.6) with (iii) in Theorem 2.3 redily yields the it (2.5).
228 Y. SHANG The following result chrcterizes the reltionship between slowly vrying functions (see e.g. [2] p. 275) nd those of α-slow increse. A function L(x) is sid to be slowly vrying if L(tx) 1, (2.7) L(t) s t, for every x > 0. An ppliction in scle-free networks cn be found in [8]. Theorem 2.5. Let C R. If is function of α-slow increse for α 1 nd f (x) is decresing, then 1, (2.8) tht is, is slowly vrying. On the other hnd, if is slowly vrying function with nd continuous derivtive f (x) > 0 nd f (x) is incresing, then is of α-slow increse for α 1. Proof. Suppose tht is of α-slow increse nd tht C > 1. Lgrnge men vlue theorem, we hve Applying the (Cx x)f (ξ) (C 1)xf (x) (C 1)xα f (x), (2.9) for some x < ξ < Cx. Combining (2.9) with Definition 1.1 gives the it (2.8). Now suppose tht C < 1. Similrly, we cn derive (x Cx)f (ξ) 1 C C 1 C C α Cxf (Cx) (Cx)α f (Cx), (2.10) for some Cx < ξ < x. Combining (2.10) with Definition 1.1 gives the it (2.8). On the other hnd, ssume tht stisfies (2.8), then by tking C > 1, we obtin (C 1)xα f (x) (C 1)xf (x) (C 1)xf (ξ) 0, (2.11) for some x < ξ < Cx nd α 1. Hence, is function of α-slow increse. Now recll well-known lemm (see e.g. [6] p. 332).
FUNCTIONS OF α-slow INCREASE 229 Lemm 2.6. If s n is sequence of positive numbers with it s, then the sequence n s1 s 2 s n hs lso it s. We conclude the pper by presenting n nlogous result for functions of α-slow increse. Theorem 2.7. If is function of α-slow increse on the intervl [, ) then the following symptotic formul holds n f()f( + 1) f(n) f(n), (2.12) where is positive number. Proof. Without loss of generlity, we ssume > 1 on the intervl [, ). Since ln is incresing nd positive, we hve by integrtion by prts ln f(i) ln dx + O(ln f(n)) i n ln f(n) From (1.1) nd the L Hôspitl rule, we derive tht nd hence If the integrl x f(t) xf (x) dx + O(ln f(n)). (2.13) ln f (x) 0, (2.14) x x ln f(n) o(n). (2.15) dt converges, we obtin x On the other hnd, if the integrl x L Hôspitl rule tht x Accordingly, from (2.16) nd (2.17) we obtin x x f(t) dt 0. (2.16) x f(t) dt diverges, we hve from (1.1) nd the f(t) dt xf (x) o(x 1 α ). (2.17) x xf (x) dx o(n1 α ). (2.18) Eqs. (2.13), (2.15) nd (2.18) imply tht ln f(i) n ln f(n) + o(n), (2.19) which is equivlent to i 1 n ln f(i) ln f(n) + o(1). (2.20) i The proof of the theorem is then complete.
230 Y. SHANG We mention tht nother generliztion of Lemm 2.6 for prime numbers is provided in the work [7]. Acknowledgments. The uthor would like to thnk the nonymous referees for creful reding nd evluting the originl version of this mnuscript. References [1] R. B. Corcino nd C. B. Corcino, On generlized Bell polynomils, Discrete Dyn. Nt. Soc. 2011 (2011) Article ID 623456. [2] W. Feller, An Introduction to Probbility Theory nd its Applictions, Volume II, John, Wiley & Sons, New York, 1970. [3] R. Jkimczuk, Functions of slow increse nd integer sequences, J. Integer Seq. 13 (2010) Article 10.1.1. [4] R. Jkimczuk, Integer sequences, functions of slow increse, nd the Bell numbers, J. Integer Seq. 14 (2011) Article 11.5.8. [5] T. Mnsour, M. Schork nd M. Shttuck, On new fmily of generlized Stirling nd Bell numbers, Electron. J. Combin. 18 (2011) #P77. [6] J. Rey Pstor, P. Pi Cllej nd C. Trejo, Análisis Mtemárico, Volumen I, Octv Edición, Editoril Kpelusz, 1969. [7] Y. Shng, On it for the product of powers of primes, Sci. Mgn 7 (2011) 31 33. [8] Y. Shng, Anlyzing the cliques in scle-free rndom grphs, J. Adv. Mth. Stud. 5 (2012) 11 18. Yilun Shng Institute for Cyber Security, University of Texs t Sn Antonio, Sn Antonio, Texs 78249, USA E-mil ddress: shylmth@hotmil.com