Certain inequalities connected with the golden ratio and the Fibonacci numbers

CERTAIN INEQUALITIES CONNECTED WITH THE GOLDEN RATIO AND THE FIBONACCI NUMBERS

Marcin Adam, Bożena Piątek, Mariusz Pleszczyński, Barbara Smoleń[*], Roman Wituła

Institute of Mathematics, Silesian University of Technology
 Gliwice, Poland
marcin.adam@polsl.pl, bozena.piatek@polsl.pl, mariusz.pleszczynski@polsl.pl, roman.witula@polsl.pl

Abstract. In the present paper we give some condensation type inequalities connected with Fibonacci numbers. Certain analytic type inequalities related to the golden ratio are also presented. All results are new and seem to be an original and attractive subject also for future research.

Keywords:  Fibonacci numbers, golden ratio, Perrin constant

1. Introduction

The notion of Fibonacci numbers and the golden ratio may be found in many branches of mathematics, including number theory, geometry, algebra, matrix theory, numerical methods, classical analysis, dynamical systems, and even spectral analysis or music (see monographs [1-3], and selective papers [4-7]). Despite such a large spread occurrence of Fibonacci numbers and the golden ratio in mathematics, still some areas of mathematics tend to be poorly represented by these objects. In our opinion, a good example of such a niche (considered also in this paper) is the area of analytic inequalities. We believe this paper opens up a new stage of discoveries, and the inequalities presented here will be classical ones in the considered area of mathematics.

The main results of the paper are presented in two sections. In the first one we investigated the condensation type inequalities associated with the Fibonacci numbers. In the second one we discuss several analytic type inequalities related to the golden ratio and Perrin constant.

2. Condensation type inequalities connected with Fibonacci numbers

We begin with the following inequality based on basic properties of the Fibonacci numbers. Let us recall that the Fibonacci numbers are defined by the following linear recurrence relation

with . As a result of the definition we get .

Theorem 1. Let and be a finite sequence of real numbers such that two inequalities are satisfied:

1.

(1)

2.

for some

(2)

Then there is an index  such that

Proof. We prove this by contradiction. Let us suppose that for each index  we have

(3)

which immediately leads to the following inequality

(4)

that can be easily shown by induction. Indeed, from (3) we have

for the initial step and

for the inductive one.

From (4), on account of (1) we obtain

Next, let us denote  where Then the left-hand side of the previous inequality is equal to the following:

Now from (2) it follows directly that  for all   Indeed,  and since , there is  So finally we obtain

which contradicts to (4). This completes the proof.

As a direct conclusion of this result we obtain the following generalization:

Corollary 2. Let  and  be a finite sequence of real numbers satisfying condition 1 of the previous theorem. If, additionally,

for some  then there is an index  such that

Remark 3. For the inequality (2) see also Chern and Cui paper [8].

3. Inequalities connected with the golden ratio

Let  denote the golden ratio, i.e.  and let .

Theorem 4. The following golden ratio type inequalities hold:

1.

for ; the equality sign is attained  for  only. The function  is increasing on interval  and decreasing on each of the intervals  and , where  (see Fig. 1). We note that

.

2.  The function

is increasing on each of the intervals  and , and decreasing on interval , where . We note that  and

.

3.  The function

is decreasing on  and increasing on .

Moreover, we have .

4.  Let us set

for , and

for . Then the function  is increasing on each of intervals  and , and decreasing on interval . On the other hand, the function  is increasing on each of the intervals  and , and decreasing on  Furthermore we obtain

Moreover, if  then (see Figs. 2-4)

and the minimum of this function is attained in , we have

Fig. 1. Plot of the function

Fig. 2. Plot of the function

Fig. 3. Plot of the function

Fig. 4. Plot of the function  in the interval (the domain of this one is equal to

Proof. We consider the following functions:

Computing derivatives of these functions we obtain

Moreover,

It is easy to check that the function  is decreasing on  and is increasing on , so we have  for  which is equivalent to the inequality  for  (the equality sign is attained here only for ). By numerical calculations, we proved that the function  is increasing on interval  and decreasing on intervals  and , where .

Similarly as , also  is decreasing on  and is increasing on , so we obtain

i.e.

We have . By numerical calculations we proved that the function  is increasing on intervals  and , and decreasing on interval . The function  is decreasing on ) and increasing on . Hence, function  has a local minimum at the point  which is equal to .

Corollary 5. By item 1, the following inequality holds

In equivalent form, we obtain

.

(5)

Proof. We have

i.e.

which implies (5) for  since  and .

Corollary 6. By item 2, the following inequality holds

Corollary 7. By item 3, the following inequality holds

More precisely, the function

is decreasing on interval  and increasing on interval .

Remark 8. Closely connected to the golden ratio is the so-called Perrin constant  defined to be the only real zero of the so-called Perrin polynomial (see [9-11])

where

and .

In relation to inequalities

,

and

from point 4 of Theorem 4, we are interested in the equivalent of these inequalities for the Perrin constant , i.e. the inequalities of the type

,

where

Since we have

so we are interested when the following system of equations hold:

(6)

which implies that  is a real solution of the following equation

i.e.

or

Finally, by numerical calculations we get precisely two triplets of real numbers being the solution of system (6):

1)  ,

2)  .

For these solutions we can deduce the following inequalities:

A) the first collection of five inequalities for the first triple (a, b, c) of solutions (see Fig. 5)

for

where

and

for

Λfor

for

for  and where the equality sign is taken only for .

The function

is increasing on , . Moreover,  is decreasing on two intervals  and , and

B)  for the second triple (a, b, c) of solutions similar inequalities can be obtained, however, due to the volume of the paper, they will be omitted.

Fig. 5. Plot of the functions  and   for the first triple  - on the left, and for the second triple  - on the right

4. Conclusions

In the paper certain inequalities connected with the golden ratio and the Fibonacci numbers are discussed. We were able to accomplish the intended overall goal of the paper, even with some excess (see in particular the results of point 4 of Theorem 4). Generalization from Remark 8 connected with the Perrin’s polynomial and constant is quite natural and in fact well-defined, but did not completely fulfill our expectations of aesthetic nature. We believe that the research subject matter indicated in this paper is still open and can encourage (especially Fibomaniacs) for active reflection.

References

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[8] Chern S., Cui A., Fibonacci numbers close to a power of 2, The Fibonacci Quarterly 2014, 52, 4, 344-348.

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[*] Currently, the fourth author, Barbara Smoleń, is a student of mathematics and this paper is a part of her Bachelor's thesis written under the supervision of Prof. Roman Wituła.