A Sharp Double Inequality for the Inverse Tangent Function
11 A Sharp Double Inequality forthe Inverse Tangent Function
Gholamreza Alirezaei
Abstract —The inverse tangent function can be bounded bydifferent inequalities, for example by Shafer’s inequality. In thispublication, we propose a new sharp double inequality, consistingof a lower and an upper bound, for the inverse tangent function.In particular, we sharpen Shafer’s inequality and calculate thebest corresponding constants. The maximum relative errors ofthe obtained bounds are approximately smaller than . and . for the lower and upper bound, respectively. Furthermore,we determine an upper bound on the relative errors of theproposed bounds in order to describe their tightness analytically.Moreover, some important properties of the obtained boundsare discussed in order to describe their behavior and achievedaccuracy. Index Terms —trigonometric bounds; Shafer’s inequality; in-verse tangent approximation;
I. I
NTRODUCTION T HE inverse tangent function is an elementary mathemati-cal function that appears in many applications, especiallyin different fields of engineering. In electrical engineering,especially in the communication theory and signal processing,it is mostly used to describe the phase of a complex-valuedsignal. But there are many other applications in which theinverse tangent function plays an important role. On the onehand, it is often used as an approximation for more complexfunctions because of its elementary behavior. For instance, theHeaviside step function is the most famous function that can bevery accurately approximated by the inverse tangent function.On the other hand, it is sometimes approximated by simplerfunctions in order to enable further calculations. For instance,the inverse tangent function can be accurately approximatedby its argument if the absolute value of the argument issufficiently small. Quite naturally the problem arises how toreplace the inverse tangent function with a surrogate function,in order to approximate the inverse tangent function as wellas other contemplable functions accurately. If a surrogatefunction with a mathematically simple form could be found,then the subsequent application of such a surrogate functionwould be considerable. A few application cases are in the fieldof information and estimation theory where an unknown phaseshift or the direction of arrival is estimated, for example byusing the CORDIC-algorithm [1], the MUSIC-algorithm [2],or MAP and ML estimators [3]. Some other cases are in thefield of system design and control theory where a non-linearnetwork unit is modeled by a non-linear function, for instance
G. Alirezaei is with the Institute for Theoretical Information Tech-nology, RWTH Aachen University, 52056 Aachen, Germany (e-mail:[email protected]).The present work is categorized in terms of Mathematics Subject Classifi-cation (MSC2010): 26D05, 26D07, 26D15, 33B10, 39B62. the saturation behavior of an amplifier [4] or the sigmoidalnon-linearity in neuronal networks [5]. Some more applicationcases are related to the theory of signals and systems, wherea signal should be mapped into a set of coefficients of basisfunctions, however the transformation is not feasible becauseof the phase description by the inverse tangent function, forexample in some Fourier-related transforms [6]. Many otherapplications are likewise conceivable.But we have to mention that finding a simple replacementfor the inverse tangent function is in fact difficult. In thepresent work, we thus focus only on a special idea whichhas some nice properties and is described in the following.In [7], R. E. Shafer proposed the elementary problem:
Showthat for all x > the inequality x √ x < arctan( x ) (1) holds , where arctan( x ) denotes the inverse tangent functionthat is defined for all real numbers x . From Shafar’s problemseveral inequalities have been emerged to date. In particular,the authors in [8] investigated double inequalities of the form a xa + √ x < arctan( x ) < b xb + √ x , x > , (2)and they determined the coefficients a , a , b and b suchthat the above double inequality is sharp. In the present work,we follow a similar idea and investigate a generalized versionof the double inequality in (2). We consider functions of thetype xc + √ c + c x (3)with positive real coefficients c , c and c , because suchkind of functions has advantageous properties in order toreplace the inverse tangent function as we will discuss inthe next section. Then we determine the triple ( c , c , c ) such that a lower and an upper bound for the inverse tangentfunction is achieved. In order to describe the tightness of theobtained bounds, we determine an upper bound on the relativeerrors of the proposed bounds. Furthermore, we discuss somecorresponding properties of the proposed bounds and visualizethe achieved results. G. A.July 18, 2013 Mathematical Notations:
Throughout this paper we denote the set of real numbersby R . The mathematical operation | x | denotes the absolutevalue of any real number x . Furthermore, O (cid:0) ω ( x ) (cid:1) denotesthe order of any function ω ( x ) . a r X i v : . [ c s . I T ] J u l II. M
AIN T HEOREMS
In the current section, we present the new bounds for theinverse tangent function and describe some of their importantproperties.
Theorem II.1
For all x ∈ R , let f ( x ) , g ( x ) and h ( x ) bedefined by f ( x ) := x π + (cid:113)(cid:0) − π (cid:1) + x π , (4) g ( x ) := arctan( x ) (5) and h ( x ) := x − π + (cid:113)(cid:0) π (cid:1) + x π . (6) Then, for all x ∈ R , the double inequality | f ( x ) | ≤ | g ( x ) | ≤ | h ( x ) | (7) holds.Proof: See Appendix A.
Remark II.2
The functions f ( x ) , g ( x ) and h ( x ) are pointsymmetric such that the identities f ( − x ) = − f ( x ) , g ( − x ) = − g ( x ) and h ( − x ) = − h ( x ) hold. Hence, it is sufficient toconsider only the case of x ≥ . Remark II.3
The triples (cid:0) π , (1 − π ) , π (cid:1) and (cid:0) − π , ( π ) , π (cid:1) are the best possible ones such that theabove double inequality holds. In other words, no componentof the first triple can be replaced by a smaller value andno component of the second triple can be replaced by alarger value with respect to x ≥ while keeping the othercomponents fixed. In this sense, the double inequality inTheorem II.1 is sharp. We get a first impression of the nature of the boundsfrom Figure 1. As we can see the double inequality inTheorem II.1 is very tight. The curves seem to be continuous,strictly increasing and convex. Hence, we elaborately discussthe mathematical properties of the obtained bounds in thefollowing.On the one hand, the first three elements in the Taylor seriesexpansions of f ( x ) , g ( x ) and h ( x ) as | x | approaches zero areobtained as f ( x ) (cid:39) x − π − x + 2 3 π − π − x + O (cid:0) x (cid:1) , (8) g ( x ) (cid:39) x − x + 15 x + O (cid:0) x (cid:1) (9)and h ( x ) (cid:39) x − x + π + 12108 x + O (cid:0) x (cid:1) . (10) Remark II.4
Only the both first elements in the Taylor seriesexpansions of f ( x ) and g ( x ) are identical to each other, whilein the Taylor series expansions of h ( x ) and g ( x ) the both x h ( x ) g ( x ) f ( x ) Fig. 1. The inverse tangent function and its bounds from Theorem II.1 arevisualized for the range of ≤ x ≤ . The curves are closely adjacent toone another such that without magnification the differences are not reallyvisible. The curves are equal at zero and approach the same upper limit as | x | approaches infinity. first two elements are pairwise identical to each other. Thus, h ( x ) achieves a better approximation of g ( x ) than f ( x ) forsufficiently small | x | . On the other hand, the first three elements in the asymptoticpower series expansions of f ( x ) , g ( x ) and h ( x ) as | x | ap-proaches infinity are obtained as f ( x ) (cid:39) π − x − π − π − πx + O (cid:0) x − (cid:1) , (11) g ( x ) (cid:39) π − x + 13 x + O (cid:0) x − (cid:1) (12)and h ( x ) (cid:39) π − π − x + π − π + 188 πx + O (cid:0) x − (cid:1) (13)by using the general definition of the asymptotic power seriesexpansion [9, p. 11, Definition 1.3.3] and simple calculations. Remark II.5
The both first two elements in the asymptoticpower series expansions of f ( x ) and g ( x ) are pairwiseidentical to each other while in the asymptotic power seriesexpansions of h ( x ) and g ( x ) only the both first elementsare identical to each other. Thus, f ( x ) achieves a betterapproximation of g ( x ) than h ( x ) for sufficiently large | x | . Corollary II.6
From equations (8) – (13) we conclude that lim x (cid:55)→± f ( x ) = lim x (cid:55)→± g ( x ) = lim x (cid:55)→± h ( x ) = 0 (14) and lim x (cid:55)→±∞ f ( x ) = lim x (cid:55)→±∞ g ( x ) = lim x (cid:55)→±∞ h ( x ) = ± π . (15) Lemma II.7
For all x ∈ R , both bounds f ( x ) and h ( x ) arecontinuous. Proof:
Both numerators and denominators of f ( x ) and h ( x ) are continuous functions in x and the denominators arealways non-zero which imply the absence of discontinuities. Lemma II.8
For all x ∈ R , both bounds f ( x ) and h ( x ) arestrictly increasing.Proof: By differentiation we obtain the following firstderivative dd x xc + √ c + c x = c + c √ c + c x √ c + c x (cid:2) c + √ c + c x (cid:3) . (16)This derivative is positive for all x ∈ R because c , c and c are positive constants in both bounds. Hence, the bounds arestrictly increasing. Corollary II.9
Both bounds f ( x ) and h ( x ) are differentiableon R , because the first derivative of the bounds exists due tothe derivative in equation (16) . Corollary II.10
Both bounds f ( x ) and h ( x ) are limited,due to equation (15) and because of the monotonicity inLemma II.8. Corollary II.11
Both bounds f ( x ) and h ( x ) do not haveany critical points, because they are strictly increasing anddifferentiable on R . Lemma II.12
For all x ≥ , both bounds f ( x ) and h ( x ) areconcave while for all x ≤ , both bounds are convex.Proof: By differentiation we obtain the following secondderivative d d x xc + √ c + c x = − c x c c + 2 c c x + 3 c √ c + c x ( c + c x ) / (cid:2) c + √ c + c x (cid:3) . (17)The sign of this derivative is only dependent on x because c , c and c are positive constants in both bounds. Hence, thisderivative is non-positive for all x ≥ and non-negative forall x ≤ which completes the proof. Corollary II.13
Both bounds f ( x ) and h ( x ) have the sameunique inflection point at the origin, due to opposing convex-ities for x ≥ and x ≤ , see Lemma II.12. In the following enumeration, we now summarize the prop-erties of the bounds that have been shown, thus far.1) The bounds are equal only at zero and they approachthe same limit as | x | approaches infinity.2) Both bounds are point symmetric, continuous, strictlyincreasing, differentiable, and limited.3) They are convex for all x ≤ and concave otherwise.4) There are no critical points.5) Both bounds have the same unique inflection point. −3 x h ( x ) − f ( x ) f ( x ) r f ( x ) r h ( x ) h ( x ) − f ( x ) h ( x )+ f ( x ) Fig. 2. The relative errors of the bounds for the inverse tangent function arevisualized for the range of ≤ x ≤ . All curves are equal at zero and theyapproach zero as | x | approaches infinity. r h ( x ) is smaller, and hence, betterthan r f ( x ) for sufficiently small values of | x | , while for sufficiently largevalues of | x | the opposite holds. The relative error r f ( x ) is approximatelysmaller than . while r h ( x ) is approximately smaller than . . Theratio [ h ( x ) − f ( x )] / [ h ( x )+ f ( x )] lies between r f ( x ) and r h ( x ) , and hence,is an upper bound for min { r f ( x ) , r h ( x ) } while [ h ( x ) − f ( x )] /f ( x ) is anupper bound for max { r f ( x ) , r h ( x ) } . The above properties enable us to use the proposed boundssuitable in future works. It remains to show the tightness ofthe bounds with respect to the inverse tangent function. Forthis purpose we deduce an upper-bound on the actual relativeerrors of the bounds, in the following.
Definition II.14
For all x ∈ R , the relative errors of thebounds given in Theorem II.1 are defined by r f ( x ) := g ( x ) − f ( x ) g ( x ) (18) and r h ( x ) := h ( x ) − g ( x ) g ( x ) . (19)Note that x = 0 is a removable singularity for both lastratios because of approximations (8), (9) and (10). Thus, r f ( x ) and r h ( x ) are continuously extendable over x = 0 .All following fractions are also continuously extendable over x = 0 in a similar manner so that no difficulties related tosingularities occur, hereinafter. Theorem II.15
For all x ∈ R , the inequalities max (cid:8) r f ( x ) , r h ( x ) (cid:9) ≤ h ( x ) − f ( x ) f ( x )= 10 − π − √ π x + (cid:113) ( π − + 4 π x π − √ π x (20) and min (cid:8) r f ( x ) , r h ( x ) (cid:9) ≤ h ( x ) − f ( x ) h ( x ) + f ( x )= 10 − π − √ π x + (cid:113) ( π − + 4 π x π − √ π x + (cid:113) ( π − + 4 π x ≤ max (cid:8) r f ( x ) , r h ( x ) (cid:9) (21) hold.Proof: See Appendix B.Note, that the inequalities in Theorem II.15 do not containthe inverse tangent function, at all.In Figure 2, the relative errors of the obtained boundsare shown. The maximum relative errors of the bounds areapproximately smaller than . and . for f ( x ) and h ( x ) , respectively. It is worthwhile mentioning that bothbounds are valid for the whole domain of real numbers.III. C ONCLUSION
In the present work, we have investigated the approximationof the inverse tangent function and deduced two new bounds.We have derived a lower and an upper bound with simpleclosed-form formulae which are sharp and very accurate.Furthermore, we have presented some useful and importantproperties of the obtained bounds. These properties can benecessary in future works. Moreover, we have investigatedthe relative errors of the proposed bounds. The correspondingmaximum relative errors of the bounds are approximatelysmaller than . and . for the lower bound andupper bound, respectively. These values show that the obtainedbounds are very accurate and thus are suitably applicable inthe most engineering problems. Finally, we have illustratedsome results in order to visualize the achieved gains.A PPENDIX AP ROOF OF THE B OUNDS
Lemma A.1
The transcendental number π can be boundedby the double inequality < π < . (22) Proof:
Both bounds are well known for long, see forexample [10]. A new proof of the upper bound can be foundin [11]. We here give an elementary proof of the lower bound.The identities (cid:80) ∞ k =1 1 k ( k +1) and ζ (2) = (cid:80) ∞ k =1 1 k = π ,see for example [12, p. 8, eq. 0.233.3 and p. 12, eq. 0.244.3],are used to deduce π ∞ (cid:88) k =1 k − ∞ (cid:88) k =1 k ( k + 1) = 1 + ∞ (cid:88) k =1 k ( k + 1)= 1 + 12 + 112 + 136 (cid:124) (cid:123)(cid:122) (cid:125) = + ∞ (cid:88) k =4 k ( k + 1) > . (23)Hence < π follows. In the following, we denote the differences g ( x ) − f ( x ) and h ( x ) − g ( x ) by ∆ f ( x ) := g ( x ) − f ( x ) (24)and ∆ h ( x ) := h ( x ) − g ( x ) , (25)respectively. From Corollary II.6 it is immediately deducedthat lim x (cid:55)→± ∆ f ( x ) = lim x (cid:55)→± ∆ h ( x )= lim x (cid:55)→±∞ ∆ f ( x ) = lim x (cid:55)→±∞ ∆ h ( x ) = 0 . (26)By direct algebra the first derivatives of ∆ f ( x ) and ∆ h ( x ) aregiven as d∆ f ( x )d x = 11 + x − π + (cid:113)(cid:0) − π (cid:1) + x π + x π (cid:113)(cid:0) − π (cid:1) + x π (cid:16) π + (cid:113)(cid:0) − π (cid:1) + x π (cid:17) (27)and d∆ h ( x )d x = −
11 + x + 11 − π + (cid:113)(cid:0) π (cid:1) + x π − x π (cid:113)(cid:0) π (cid:1) + x π (cid:16) − π + (cid:113)(cid:0) π (cid:1) + x π (cid:17) , (28)respectively. Corollary A.2
The first derivatives of ∆ f ( x ) and ∆ h ( x ) vanish only at three real points, namely x f ∈ (cid:26) , ± ( π − √− π + 36 π − π − π − (cid:27) (29) and x h ∈ (cid:26) , ± √− π + 108 π − π (10 − π ) (cid:27) , (30) respectively.Proof: We set (27) and (28) equal to zero and obtain thepoints in (29) and (30) by direct calculations. It remains toprove that all points are real. This is done by showing that thediscriminant functions y f ( ν ) := − ν + 36 ν −
160 = 2( ν − − ν ) (31)and y h ( ν ) := − ν + 108 ν −
576 = (5 ν − − ν ) (32)are non-negative for ν = π . A curve tracing of y f ( ν ) and y h ( ν ) leads to the relationships y f ( ν ) ≥ ⇔ ≤ ν ≤ (33)and y h ( ν ) ≥ ⇔ ≤ ν ≤ , (34) respectively. Hence, both y f ( ν ) and y h ( ν ) are non-negative forall ≤ ν ≤ . By comparing the latter double inequalitywith the double inequality in Lemma A.1 we deduce that y f ( π ) and y h ( π ) are non-negative, and hence, all roots in (29)and (30) are real. Corollary A.3
The difference ∆ f ( x ) is positive for all suf-ficiently small positive real numbers x . For all negative realnumbers x with sufficiently small absolute value, the difference ∆ f ( x ) is negative.Proof: We incorporate the equations (8) and (9) into (24)to derive the first-order approximation of ∆ f ( x ) as ∆ f ( x ) (cid:39) − π π − x + O (cid:0) x (cid:1) (35)for all sufficiently small values of | x | . From the doubleinequality in Lemma A.1, we deduce that the last ratio isalways positive which completes the proof. Corollary A.4
The difference ∆ h ( x ) is positive for all suf-ficiently large positive real numbers x . For all negative realnumbers x with sufficiently large absolute value, the difference ∆ h ( x ) is negative.Proof: We incorporate the equations (12) and (13)into (25) to derive the first-order asymptotic approximationof ∆ h ( x ) as ∆ h ( x ) (cid:39) − π x − + O (cid:0) x − (cid:1) (36)for all sufficiently large values of | x | . From the doubleinequality in Lemma A.1, we deduce that the last ratio isalways positive which completes the proof. Proof of T HEOREM
II.1:
We only consider the case of x ≥ . The case of x ≤ can be proved analogously, due tothe point symmetric property of all functions in Theorem II.1.On the one hand, we know from Corollary A.2 that each ofdifferences ∆ f ( x ) and ∆ h ( x ) has only one stationary point forall x > . On the other hand, each of them attains equal valuesat x = 0 and as x (cid:55)→ ∞ , i.e., ∆ f (0) = ∆ f ( x (cid:55)→ ∞ ) = 0 and ∆ h (0) = ∆ h ( x (cid:55)→ ∞ ) = 0 , according to the equation (26).Hence, and because of Corollary A.3 and A.4, as x increasesfrom zero to infinity each of the differences ∆ f ( x ) and ∆ h ( x ) increases monotonically from zero to a maximum value andfrom there on decreases monotonically toward zero. Thus, bothdifferences ∆ f ( x ) and ∆ h ( x ) are non-negative for all x ≥ .In other words, if one of the differences had at least one signchange for some value of x > , then it would have at leasttwo stationary points for x > , but this contradicts the curvetracing in Corollary A.2.A PPENDIX BP ROOF OF THE R ELATIVE E RRORS
Definition B.1
Let r f ( x ) and r h ( x ) be defined as in Defini-tion II.14. Then, we define two auxiliary sets by I f := (cid:8) x ∈ R | r f ( x ) ≥ r h ( x ) (cid:9) (37) and I h := (cid:8) x ∈ R | r f ( x ) < r h ( x ) (cid:9) . (38)Note that both sets I f and I h are disjoint and their union isthe whole real domain. Corollary B.2
For all x ∈ I f , x ≥ , the inequality g ( x ) ≥ h ( x ) + f ( x )2 (39) holds. If x ∈ I h , x ≥ , then the inequality g ( x ) < h ( x ) + f ( x )2 (40) holds. In the case of x ∈ I f with x < and x ∈ I h with x < the above inequalities are reversed.Proof: For all x ∈ I f with x ≥ , and from Defini-tion II.14 and B.1 it follows that r f ( x ) ≥ r h ( x ) ⇔ r f ( x ) g ( x ) ≥ r h ( x ) g ( x ) ⇔ g ( x ) − f ( x ) ≥ h ( x ) − g ( x ) ⇔ g ( x ) ≥ h ( x ) + f ( x )2 . (41)Similarly, for all x ∈ I h with x ≥ it follows that r f ( x ) < r h ( x ) ⇔ r f ( x ) g ( x ) < r h ( x ) g ( x ) ⇔ g ( x ) − f ( x ) < h ( x ) − g ( x ) ⇔ g ( x ) < h ( x ) + f ( x )2 . (42)In the case of x < , the functions f ( x ) , g ( x ) and h ( x ) arenegative, and hence, the inequalities in (39) and (40) arereversed. Proof of T HEOREM
II.15:
The proof of inequality (20)follows from inequality (7) and Definition II.14. It gives r f ( x ) = g ( x ) − f ( x ) g ( x ) ≤ h ( x ) − f ( x ) g ( x ) ≤ h ( x ) − f ( x ) f ( x ) (43)and r h ( x ) = h ( x ) − g ( x ) g ( x ) ≤ h ( x ) − f ( x ) g ( x ) ≤ h ( x ) − f ( x ) f ( x ) (44)which in turn result in max (cid:8) r f ( x ) , r h ( x ) (cid:9) ≤ h ( x ) − f ( x ) f ( x ) . (45)The proof of inequality (21) follows from Definition II.14 andCorollary B.2. For all x ∈ I f with x ≥ , it gives r f ( x ) = 1 − f ( x ) g ( x ) ≥ − f ( x ) h ( x )+ f ( x )2 = h ( x ) − f ( x ) h ( x ) + f ( x ) (46)and r h ( x ) = h ( x ) g ( x ) − ≤ h ( x ) h ( x )+ f ( x )2 − h ( x ) − f ( x ) h ( x ) + f ( x ) (47)which in turn result in r h ( x ) ≤ h ( x ) − f ( x ) h ( x ) + f ( x ) ≤ r f ( x ) . (48) If x ∈ I h with x ≥ , then it gives r f ( x ) = 1 − f ( x ) g ( x ) < − f ( x ) h ( x )+ f ( x )2 = h ( x ) − f ( x ) h ( x ) + f ( x ) (49)and r h ( x ) = h ( x ) g ( x ) − > h ( x ) h ( x )+ f ( x )2 − h ( x ) − f ( x ) h ( x ) + f ( x ) (50)which in turn result in r f ( x ) < h ( x ) − f ( x ) h ( x ) + f ( x ) < r h ( x ) . (51)From the double inequalities (48) and (51), we deduce that min (cid:8) r f ( x ) , r h ( x ) (cid:9) ≤ h ( x ) − f ( x ) h ( x ) + f ( x ) ≤ max (cid:8) r f ( x ) , r h ( x ) (cid:9) (52)for all x ≥ . For the case of x < , the proof can beobtained analogously. The identities in (20) and (21) arise fromstraightforward calculations.A CKNOWLEDGMENT
Research described in the present work was supervised byUniv.-Prof. Dr. rer. nat. R. Mathar, Institute for TheoreticalInformation Technology, RWTH Aachen University. The au-thor would like to thank him for his professional advice andpatience. R
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