LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS VIA TAYLOR S EXPANSION WITH INTEGRAL REMAINDER S. S. DRAGOMIR ;2 Astrct. In this pper we otin severl new logrithmic inequlities for two numers ; minly in the cse when > 0 y the use of Tylor s epnsion with integrl reminder. An nlysis of which ound is etter is lso performed.. Introduction There re numer of inequlities for logrithm tht re well know nd widely used in literture, such s (.) ln for > 0; (.2) nd see for instnce 2 2 + ln ( + ) p for 0; + ln ( ) ; for < ; ln n n for n > 0 nd > 0; ln ( jj) ln ( + ) ln ( jj) for jj < ; 3 2 ln ( ) 3 for 0 < 05838; 2 http functionswolfrmcomelementryfunctionslog29 nd [4]. A simple proof of the rst inequlity in (.2) my e found, for instnce, in [5], see lso [6] where the following rtionl ounds re provided s well + 5 6 ( + ) + 3 ln ( + ) + 6 + 2 3 for 0 (.3) In the recent pper [] we estlished the following inequlities for logrithm ( ) 2n 2n 2n m 2n f; g X ( ) k ( ) k k k k ln + ln 99 Mthemtics Suject Clssi ction. 26D5, 26D0. Key words nd phrses. Logrithmic inequlities, Tylor s epnsion, Bounds. ( ) 2n 2n min 2n f; g
2 S. S. DRAGOMIR ;2 nd (.4) ( ) 2n 2n 2n m 2n f; g ln ln X k ( ) k k k for ny ; > 0 nd for n. In prticulr, for n we get (.5) nd (.6) ( ) 2 2 m 2 f; g ln + ln ( ) 2n 2n min 2n f; g ( )2 2 min 2 f; g ( ) 2 2 m 2 f; g ln ln ( )2 2 min 2 f; g We hve the following upper ounds [] 2n X ( ) k ( ) k (.7) (0 ) k k ln + ln ( )2n 2n 2n (2n ) 2n 2n nd k (0 ) ln ln 2n X k ( ) k k k ( )2n 2n 2n (2n ) 2n 2n for ny ; > 0 nd for n. For ny ; > 0 we hve the simpler inequlities (.8) (0 ) nd (.9) (0 ) ln ln ln + ln We lso hve the Hölder s type upper ounds [2] (.0) nd (.) 2n X (0 ) k ( ) k ( ) k k k ( )2 ( )2 ln + ln j j 2n +p 2nq 2nq q [(2n ) p + ] p (2nq ) q 2n q () (0 ) ln ln 2n X k ( ) k k k j j 2n +p 2nq 2nq q ; [(2n ) p + ] p (2nq ) q 2n q () where p; q > with p + q nd ; > 0 nd n. In prticulr, we get for n (.2) (0 ) ln + ln j j+p 2q 2q q [(p + )] p (2q ) q () 2 q
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS 3 nd (.3) (0 ) ln ln j j+p 2q 2q q [(p + )] p (2q ) q () 2 q In [2] we otined the following complementry results (.4) nd (.5) 2 m f; g ( )2 ln ln 2 m f; g ( )2 ( )2 2 min f; g ln + ln for ny ; > 0. If n ; then for ny ; > 0 we hve tht [2] (.6) nd (.7) ( ) 2n+2 (2n + ) (2n + 2) m 2n+ f; g ln ln + 2n+ X ( ) 2n+2 (2n + ) (2n + 2) min 2n+ f; g ( ) 2n+2 (2n + ) (2n + 2) m 2n+ f; g ( )2 2 min f; g ( ) k ( ) k k (k ) k 2n+ X ( ) k k (k ) k ln + ln ( ) 2n+2 (2n + ) (2n + 2) min 2n+ f; g For other similr ounds see [2]. Motivted y the ove results, we estlish in this pper severl ounds for the quntities nd ln + ln ln ln k ( ) k ( ) k k k ; k ( ) k k k ; ( ) k ( ) k k (k ) k ln + ln ( ) k k (k ) k ln + ln where ; > 0 nd n The simpler cses when n re outlined nd in this cse some ounds re numericlly compred to conclude tht neither is lwys est.
4 S. S. DRAGOMIR ;2 2. Some Inequlities for Logrithm The following result holds, see for instnce [3] where vrious pplictions in Informtion Theory were provided Lemm. For ny ; > 0 we hve for m tht mx ( ) k ( ) k (2.) ln ln + k k ( ) m k Z ( t) m t m+ dt For recent inequlities derived from this identity nd short proof, see []. We hve the following result Theorem. For ny ; > 0 we hve for n tht (2.2) nd (2.3) (2n + ) 2n+ ( )2n+ ln ln (2n + ) 2n+ ( )2n+ ln ln k ( ) k ( ) k k k ( )2n+ (2n + ) 2n+ k ( ) k k k (2n + ) 2n+ ( )2n+ Proof. If we tke m 2n with n in (2.), then we get for ny ; > 0 tht (2.4) ln ln + If > > 0; then we hve nd since 2n+ Z k ( t) 2n dt Z ( ) k ( ) k k k Z ( t) 2n dt we get, y (2.4) the desired inequlity (2.2). If > > 0; then (2.5) We hve (2.6) Oserve tht Z 2n+ Z Z ( t) 2n dt ( t) 2n t 2n+ dt ( t) 2n dt Z Z (t ) 2n dt Z ( t) 2n t 2n+ dt 2n+ 2n + ( Z ( t) 2n t 2n+ dt )2n+ ( t) 2n t 2n+ dt Z ( t) 2n t 2n+ dt 2n+ ( )2n+ 2n + ( t) 2n dt Z ( t) 2n dt ( )2n+ 2n +
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS 5 nd y (2.6) we then get tht is equivlent to ( ) 2n+ dt (2n + ) 2n+ ( ) 2n+ 2n+ (2n + ) Z Z By using (2.4) nd (2.5) we get (2.2) gin. Now, if we replce with in (2.2) we get nmely ( t) 2n t 2n+ dt (2n + ) 2n+ ( )2n+ ln ln + ( ) 2n+ 2n+ (2n + ) ( t) 2n ( )2n+ t 2n+ dt dt (2n + ) 2n+ k ( ) k ( ) k k k ( )2n+ (2n + ) 2n+ (2n + ) 2n+ ( )2n+ ln ln + k ( ) k k k (2n + ) 2n+ ( )2n+ If we multiply this inequlity y we get the desired inequlity (2.3). (2.7) nd (2.8) For ny ; > 0 we hve y (2.2) nd (2.3) for n tht 3 3 ( )3 ln ln 3 3 ( )3 ln ln ( )2 + 2 2 ( )3 33 ( ) 2 2 2 3 3 ( )3 Corollry. For ny ; > 0 with > 0 we hve for n tht (2.9) nd (2.0) k ( ) k ( ) k k k k ln ln ( ) k For ny ; > 0 with > 0 we hve (2.) nd (2.2) ( ) 2 k k ln ln 2 2 ln ln + ( )2 2 2 ln ln
6 S. S. DRAGOMIR ;2 Remrk. If we tke 2 (0; ) nd in (2.2) nd (2.3), then we get (2.3) nd (2.4) ( ) 2n+ (2n + ) 2n+ ln k (2n + ) 2n+ ( )2n+ ln for ny 2 (0; ) nd n In prticulr, for n we hve (2.5) nd (2.6) ( ) k ( ) k ( )2n+ k (2n + ) k ( ) k k k 3 3 ( )3 ln + + 2 ( )2 ( )3 3 3 3 ( )3 ln Now, if ; then we get from (2.3) tht (2.7) nd from (2.4) tht (2.8) We hve k ( ) 2 ( ) k ( ) k ln k k ( ) k k k ln 2 2 3 ( )3 Theorem 2. For ny ; > 0 with > 0 we hve for n tht (2.9) (0 ) ln ln k ( )2n+ (2n + ) ( ) k ( ) k k k ( ) 2n 2n 2n 2n 2n 2n If p; q > with p + q nd ; > 0 with > 0 we hve for n tht (2.20) (0 ) ln ln k ( ) k ( ) k k k (2np + ) p [(2n + ) q ] q ( )2n+p (2n+)q (2n+)q q 2n+p 2n+p In prticulr, for p q 2 we hve (2.2) (0 ) ln ln k ( ) k ( ) k k k ( ) 2n+2 4n+ 4n+ 2 (4n + ) 2n+2 2n+2
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS 7 Proof. By (2.4) we hve for > 0 tht (2.22) ln ln k ( ) k ( ) k k k Z ( ( t) 2n t 2n+ dt ) 2n Z t 2n+ dt ( ) 2n 2n 2n 2n 2n 2n tht proves the inequlity (2.9). Using Hölder s integrl inequlity we hve for p; q > with p + q tht Z Oserve tht ( t) 2n t 2n+ dt Z! p Z q ( t) 2np dt t dt! (2n+)q nd Z Z ( t) 2np dt! p! p ( )2np+ ( )2n+p 2np + (2np + ) p q t dt! (2n+)q (2n+)q+ (2n+)q+ q (2n + ) q + (2n+)q (2n+)q [(2n + ) q ] (2n+)q (2n+)q (2n+)q (2n+)q q [(2n + ) q ] q 2n+ q 2n+ q (2n+)q (2n+)q q [(2n + ) q ] q 2n+p 2n+p By utilising the equlity in (2.22) we deduce the desired result (2.20). q For ny ; > 0 with > 0 we hve y (2.9), (2.20) nd (2.2) for n tht ( )2 (2.23) (0 ) ln ln + 2 2 ( ) 2 2 2 2 2 2 ; (2.24) (0 ) ln ln + ( )2 2 2 ( ) 2+p 3q 3q q (2p + ) p (3q ) q 2+p 2+p with p; q > with p + q nd (2.25) (0 ) ln ln + ( )2 2 2 ( ) 52 5 5 2 5 52 52
8 S. S. DRAGOMIR ;2 If ; then we hve for n tht (2.26) (0 ) ln k ( ) k ( ) k k ( ) 2n 2n 2n 2n If p; q > with p + q nd ; then we hve for n tht (2.27) (0 ) ln k ( ) k ( ) k In prticulr, for p q 2 we hve (2.28) (0 ) ln for ny k k ( ) k ( ) k k (2np + ) p [(2n + ) q ] q ( )2n+p (2n+)q q 2n+p ( ) 2n+2 4n+ 2 (4n + ) 2n+2 3. Further Inequlities for Logrithm We hve the following representtion result [2] Lemm 2. For ny m 2 nd ny ; > 0 we hve (3.) ln ln + mx ( ) k ( ) k k (k ) k ( )m m We hve Theorem 3. For ny ; > 0 we hve for n tht (3.2) nd (3.3) ( ) 2n+ 2n (2n + ) 2n+ + ( ) 2n+ 2n (2n + ) 2n ( ) 2n+ 2n (2n + ) 2n ( ) 2n+ 2n (2n + ) 2n+ Proof. If we tke m 2n with n in (3.), then we get ln ln tht is equivlent to (3.4) for ny ; > 0 + ( ) k ( ) k k (k ) k Z ( t) m dt t m ( ) k ( ) k k (k ) k ln + ln ( ) k k (k ) k ln + ln 2n ( ) k ( ) k k (k ) k ln + ln + 2n Z Z ( t) 2n dt t 2n ( t) 2n dt t 2n
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS 9 If > > 0; then we hve 2n Z ( t) 2n dt Z ( t) 2n t 2n dt 2n nmely (3.5) (2n + ) 2n ( Z ( t) 2n )2n+ t 2n If > > 0; then Z ( t) 2n Z t 2n dt Oserve tht We hve Z ( t) 2n dt 2n Z Z (t ) 2n dt (t ) 2n dt Z dt Z ( t) 2n dt (2n + ) 2n ( )2n+ ( t) 2n dt t 2n ( )2n+ 2n + ( t) 2n t 2n dt 2n nmely ( ) 2n+ Z ( t) 2n 2n 2n + t 2n dt which, y multiplying with gives (3.6) ( ) 2n+ 2n 2n + Z Z ( )2n+ 2n + (t ) 2n dt ( ) 2n+ 2n ; 2n + ( t) 2n t 2n dt ( ) 2n+ 2n 2n + for > 0 Using the representtion (3.4) nd the inequlities (3.5) nd (3.6) we get (3.2). If we replce with in (3.2) then we get nmely (3.7) ( ) 2n+ 2n (2n + ) 2n+ ( ) 2n+ 2n (2n + ) 2n+ ( ) k ( ) k k (k ) k ln + ln + ( ) 2n+ 2n (2n + ) 2n ( ) k k (k ) k ln + ln + 2n (2n + ) ( ) 2n+ 2n If we multiply (3.7) y ; then we get (3.3). (3.8) nd (3.9) For ny ; > 0 we hve y (3.2) nd (3.3) for n tht ( ) 3 6 3 + ( ) 2 2 ( ) 3 6 2 ( ) 2 2 ln + ln ( ) 3 6 2 ln + ln ( ) 3 6 3
0 S. S. DRAGOMIR ;2 We hve Corollry 2. For ny ; > 0 with > 0 we hve for n tht (3.0) ln ln + ( ) k ( ) k k (k ) k nd (3.) ln ln ( ) k k (k ) k For ny ; > 0 with > 0 we hve (3.2) ln ln nd (3.3) ln ln ( ) 2 2 + ( ) 2 2 Remrk 2. If we tke 2 (0; ) nd in (3.2) nd (3.3), then we get (3.4) nd (3.5) ( ) 2n+ 2n (2n + ) 2n+ + ( ) 2n+ 2n (2n + ) 2n In prticulr, for n we hve (3.6) nd ( ) 3 6 3 + ( ) 2 2 (3.7) ( ) 3 6 2 for ny > 0 For we lso hve (3.8) nd (3.9) ln for ny n ln ( ) 2n+ 2n (2n + ) ( ) k k (k ) ( )k ln ( ) k k (k ) k ln 2n (2n + ) ( )2n+ ( ) 2 2 + ln ( ) 3 6 ln ( )3 6 ( ) k ( )k k (k ) ( ) k k (k ) k ( )
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS In prticulr, we hve (3.20) nd (3.2) ln for ny We hve ln ( ) 2 2 + ( ) 2 2 ( ) Theorem 4. For ny ; > 0 with > 0 we hve for n tht (3.22) (0 ) + ( ) k ( ) k k (k ) k ln + ln ( ) 2n 2n 2n 2n (2n ) 2n 2n If p; q > with p + q nd ; > 0 with > 0 we hve for n tht (3.23) (0 ) + ( ) k ( ) k k (k ) k ln + ln ( ) 2n+p 2nq 2nq q 2n (2np + ) p (2nq ) q 2n+ q 2n q In prticulr, for p q 2 we hve (3.24) (0 ) + ( ) k ( ) k k (k ) k ln + ln ( ) 2n+2 4n 4n 2 2n (4n + ) 2 (4n ) 2 2n+2 2n 2 Proof. For ny ; > 0 with > 0 we hve for n tht (3.25) which proves (3.22). 2n ( ) k k (k ) Z 2n ( 2n ( ( ) k k ln + ln + ( t) 2n dt t 2n Z )2n )2n t 2n dt 2n+ 2n + 2n+ 2n + ( ) 2n 2n 2n 2n (2n ) 2n 2n ;
2 S. S. DRAGOMIR ;2 Using Hölder s integrl inequlity we hve for p; q > with p + q tht Z ( t) 2n dt t 2n Z! p Z ( t) 2np dt t 2nq dt! q ( )2np+ 2np + ( )2n+p (2np + ) p! p 2nq+ 2nq+ 2nq + 2nq + 2nq 2nq q (2nq ) 2nq 2nq ( )2n+p 2nq 2nq (2np + ) p (2nq ) q 2n q for ny ; > 0 with > 0 nd n Using the rst identity in (3.25) we get (3.23). q 2n q q For ny ; > 0 with > 0 we hve y (3.22)-(3.24) for n tht (3.26) (0 ) (3.27) nd (0 ) (3.28) (0 ) + ( ) 2 2 + ( ) 2 2 ln + ln ( ) 3 2 2 ; ln + ln ( ) 2+p 2q 2q q 2 (2p + ) p (2q ) q 3 q 2 q + ( ) 2 2 Remrk 3. For we lso hve (3.29) (0 ) + ln + ln 2 p 5 ( ) k k (k ) ( )k ln ( ) 2n 2n 2n (2n ) 2n ; r ( ) 3 2 + + 2 2 (3.30) nd (3.3) (0 ) + ( ) k k (k ) ( )k ln ( ) 2n (2np + ) p (2nq ) q (0 ) + ( ) k k (k ) ( )k ln 2n+p 2nq q 2n+ q 2n (4n + ) 2 (4n ) 2 ( ) 2n+2 4n 2 2n+2
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS 3 In prticulr, for n ; we hve (3.32) (0 ) (3.33) nd (3.34) for ny (0 ) + ( ) 2 2 + ( ) 2 2 ln ln ( ) 3 2 2 ; ( ) 2+p 2q q 2 (2p + ) p (2q ) q 3 q (0 ) 2 p 5 + ( ) 2 2 ln r ( ) 3 2 + + 2 4. Comprison Using the inequlities (2.7), (2.23) nd (2.25) we hve for ny > 0 the following upper ounds for the positive quntity (4.) ln ln where + ( )2 2 2 B (; ) ; B 2 (; ) nd B 3 (; ) (4.2) B (; ) 3 3 ( )3 ; B 2 (; ) ( ) 3 ( + ) 2 2 2 nd (4.3) B 3 (; ) r ( ) 3 4 + 3 + 2 2 + 3 + 4 5 2 2 Consider the simpler quntities C (; ) 3 ; C 2 (; ) + 2 2 nd C 3 (; ) r 4 + 3 + 2 2 + 3 + 4 5 2 If we plot the di erence D (; ) C (; ) C 2 (; ) on the domin 0 we hve the Figure, the plot of the di erence D 2 (; ) C 2 (; ) C 3 (; ) on the domin 0 2 is depicted in Figure 2 while the plot of the di erence D 3 (; ) C (; ) C 3 (; ) on the tringle 0 is incorported in Figure 3 The plots in Figure -3 show tht neither of the upper ounds B (; ) ; B 2 (; ) nd B 3 (; ) for the positive quntity is lwys est. ln ln + ( )2 2 2 ; 0 <
4 S. S. DRAGOMIR ;2 Figure. Plot of D (; ) for 0 Figure 2. Plot of D 2 (; ) for 0 2 From the inequlities (3.8), (3.26) nd (3.28) we hve for ny > 0 the following upper ounds for the positive quntity (4.4) where + ( ) 2 2 ln + ln K (; ) ; K 2 (; ) nd K 3 (; ) (4.5) K (; ) ( ) 3 6 2 ; K 2 (; ) ( ) 3 2 2
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS 5 Figure 3. Plot of D 3 (; ) for 0 nd (4.6) K 3 (; ) r 2 p ( ) 3 2 + + 2 5 2 The interested reder my perform similr nlysis for these ounds. However the detils re not provided here. References [] S. S. Drgomir, Some inequlities for logrithm vi Tylor s epnsion with integrl reminder, RGMIA Res. Rep. Coll. 9 (206), Art. 08. [http//rgmi.org/ppers/v9/v908.pdf]. [2] S. S. Drgomir, New inequlities for logrithm vi Tylor s epnsion with integrl reminder, RGMIA Res. Rep. Coll. 9 (206), Art. 26. [http//rgmi.org/ppers/v9/v926.pdf]. [3] S. S. Drgomir nd V. Glušµcević, New estimtes of the Kullck-Leiler distnce nd pplictions, in Inequlity Theory nd Applictions, Volume, Eds. Y. J. Cho, J. K. Kim nd S. S. Drgomir, Nov Science Pulishers, New York, 200, pp. 23-37. Preprint in Inequlities for Csiszár f-divergence in Informtion Theory, S.S. Drgomir (Ed.), RGMIA Monogrphs, Victori University, 200. [ONLINE http//rgmi.vu.edu.u/monogrphs]. [4] L. Kozm, Useful inequlities chet sheet, http//www.lkozm.net/inequlities_chet_sheet/. [5] E. R. Love, Some logrithm inequlities, The Mthemticl Gzette, Vol. 64, No. 427 (Mr., 980), pp. 55-57. [6] E. R. Love, Those logrithm inequlities!, The Mthemticl Gzette, Vol. 67, No. 439 (Mr., 983), pp. 54-56. Mthemtics, College of Engineering & Science, Victori University, PO Bo 4428, Melourne City, MC 800, Austrli. E-mil ddress sever.drgomir@vu.edu.u URL http//rgmi.org/drgomir 2 School of Computer Science & Applied Mthemtics, University of the Witwtersrnd, Privte Bg 3, Johnnesurg 2050, South Afric