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INEQUALITIES AMONG RELATED PAIRS OF FIBONACCI NUMBERS
K. T. Atanassov
CLBME-Bulgarian Academy of Sciences, PO Box 12, Sofia-1113, Bulgaria
Ron Knott
RDK Internet Consultancy, 1 Upperton Road, Guildford, GU2 7NB, UK
Kiyota OzeM
Utsunomiya University, 350 Mine-machi, Utsunomiya City, Japan
A. G. Shannon
KvB Institute of Technology, North Sydney, 2060, and
Warrane College, The University of New South Wales, 2033, Australia
Laszlo Szalay
University of West Hungary, H-9400, Sopron, Bajcsy Zs, ut 4, Hungary
(Submitted November 2000-Finai Revision January 2001)
1. INTRODUCTION
What may be called "Fibonacci inequalities" have been studied in a variety of contexts. These
include metric spaces [6], diophantine approximation [7], fractional bounds [5], Fibonacci numbers of graphs [1], and Farey-Fibonacci fractions [2]. Here we consider Fibonacci inequalities.
The results are the "best possible" and we relate them through the sequence {fnr}"=0 defined by
m
r
= F F
r n+l-r>
where n is a fixed natural number and FhF2,F3,..., are the ordinary Fibonacci numbers.
2. MAM RESULT
Theorem: For every natural number k, the following inequalities for the elements of the sequence
{mk}"km0 are valid:
(a) if T? = 4&, then
F F
l 4k
> F3F4k-2
> " ' ' > F2k-lF2k+2
> F2kF2k+l
> F2k-2F2k+3
> ' *' >
F F
2 4k-li
(b) if/i = 4* + l,then
F F
l 4k+l
> F3F4k~l
F
2k-lF2k+3
>'••>
> F2kUF2k-hl
> F2kF2k+2
>"*>
F F
2 4k'>
(e) if?? = 4Jfc+2,then
F F
l 4k+2
>
F F
3 4k
> " *8 > F2k+lF2k+2
> F2kF2k+3
> F2k-2F2k+5
>°'°>
F F
2 4k+\l
(d) iff? = 4A + 3? then
F F
l 4k+3
> F3F4kU
>'">
F
2k+lF2k+3
> F2k-l-2F2k+2
> F2kF2M
> ''' > F2F4k+2
•
Examples when k = 3:
(a) FxFl2 = 144>F3FlQ = 110>F5FB = 105>F6F7 = 104>F4F9 = 102>F2Fn = 89;
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INEQUALITIES AMONG RELATED PAIRS OF FIBONACCI NUMBERS
(b) FxFl3 = 233 > F3Fn = 178 > F5F9 = 170 > F7F7 = 169 > F6F8 = 168
> F4F10 = 165 >F2F12 = 144;
(c) FXFX4 = 377 > F3Fl2 = 288 > F7FK = 273 > F6F9 = 272 > F4Fn = 267 > F2F13 = 233;
(i) FtFl5 = 610 > F3Fl3 = 466 > F7F9 = 442 > F8F8 = 441 > F6F10 = 440
> F4Fl2 = 432 > F2Fl4 = 377.
Proof of Theorem: We shall use Induction simultaneously on «, that Is, on k and i.
For k = 1, we have
F1F4 = l x 3 > 2 x l = F3F2?
F ^ = 1 x 5 > 2 x 2 = F3F3 > 1 x 3 = F2 F4,
F ^ = 1 x 8 >2 x 3 = F3F4 > 1 x 5 = F2F5,
FXF7 = 1 x 13 > 2 x 5 = F3F5 > 3 x 3 = F4F4 > 1 x 8 = F2F5.
Assume that inequalities (a), (b), (c), and (d) are true for some k>\. Then we must prove that
the inequalities are true for k +1.
In particular, from the truth of (d), it follows that, for every i, \<i<k:
We shall discuss case (a), but the other cases are proved similarly.
First, we see that
FxF4k+4 - F3F4k+2 = F4k+3 + F4k+2 - 2F4k+2 - F4k+3 - F4k+2 > 0,
that is, the inequality
F2i-iF4k_2j+6 > F2i+lF4k_2i+4
is validfori = 1.
Let us assume that, for some i, \<i<ky
the inequality
(2.2)
Inequality (2.2) is true. Then we must prove that
•^•+1^4Jfc-2f+4
>
^+3^fc-2/+2
(2.3)
is also true.
But
-^•+l-MJfc-2i+4 "" ^ / + 3 ^ 4 J t - 2 i + 2
=
^+l-MJfc-2i+3
+
-^j+1^4&-2Z+2 ~ ^2i+V4k-2i+2
~ ^/+l-Mfc-2i+3 ~ ^/+2^fc-2i+2
>
~
^2i+2^4k-2i+2
®
because of the Inductive assumption for (d) In case (2.1). Therefore, Inequality (3) holds, from
which the truth of (a) follows.
3. DISCUSSION
By analogy with the extremal problems discussed In [3], we can formulate the following.
Corollary: For every natural number n the maximal element of the sequence {%}^=0 is FxFn.
Proof of Corollary: Equation (I26) of [8] can be rewritten is
Fn = Fk+lFn_k +FkFn_k_l
to show that Fn > Fn_kFku, \<k<n,
2003]
which gives the required result.
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INEQUALITIES AMONG RELATED PAIRS OF FIBONACCI NUMBERS
More generally, FxFn is the maximal element of the set
M={EE,...,E
E}9
where n = ix + ?2 + • • • + ik, 1 < A: < w, is a positive partition into k parts. The result can be proved
by induction.
4. CONCLUDING COMMENTS
A somewhat analogous result was proposed by Bakinova [4] and proved by Mascioni [9]:
F\
F
3
F5
F6
E4
F2
The results can be generalized to the sequence {un(0,1; p, q)} defined by the recurrence
with UQ = 0, ux - 1, p * 0, /? G Z, A = /?2 + 4g > 0, in which case a neater proof comes from the
use of the Binet forms for the general terms and hyperbolic cosines and sines. The result can also
be extended to related triples of Fibonacci numbers.
ACKNOWLEDGMENT
Appreciation is expressed for comments by an anonymous referee and for references suggested by the editor.
REFERENCES
A. F. Alameddine. "Bounds on the Fibonacci Number of a Maximal Outerplanar Graph." The
Fibonacci Quarterly 36.3 (1998):206-10.
K. Alladi & A. G. Shannon. "On a Property of Farey-Fibonacci Fractions." The Fibonacci
Quarterly 15.2 (1977): 153-55.
K. T. Atanassov. "One Extremal Problem." Bulletin of Number Theory & Related Topics 8.3
(1984):6-12.
V. Bakinova. "Problem B-486." The Fibonacci Quarterly 2@A (1982):366.
J. H. E. Cohn. "Recurrent Sequences Including N." The Fibonacci Quarterly 29.1 (1991):
30-36.
M. Deza & M. Laurent. "The Fibonacci and Parachute Inequalities for /i-metrics." The Fibonacci Quarterly 30,1 (1992):54-61.
C. Eisner. "On Diophantine Approximations with Rationals Restricted by Arithmetical Conditions." The Fibonacci Quarterly 38.1 (2000):25-34.
V. E.-Hoggatt, Jr. Fibonacci and Lucas Numbers, p. 59. Boston: Houghton-Mifflin, 1969.
V. D. Mascioni. "Solution to Problem B-486." The Fibonacci Quarterly 21.4 (1983):30910.
AMS Classification Number: 11B39
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