Certain inequalities connected with the golden ratio and the Fibonacci numbers
Journal of Applied Mathematics and Computational Mechanics
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
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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
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:
Then there is an index such that
Proof. We prove this by contradiction. Let us suppose that for each index we have
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 .
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:
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
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
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
Proof. We have
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])
In relation to inequalities
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
Since we have
so we are interested when the following system of equations hold:
which implies that is a real solution of the following equation
Finally, by numerical calculations we get precisely two triplets of real numbers being the solution of system (6):
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 and where the equality sign is taken only for .
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
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.
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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.