**Published in:**Journal of Scientometric Research, 14 September 2023; 12(2): 229-233.

**Published online:**14 September 2023

**DOI:**10.5530/jscires.12.2.022

## ABSTRACT

The Hirsch function, denoted as h_{f}, of a given continuous function f is a new function depending on a parameter. It exists provided some assumptions are satisfied. If this parameter takes the value one, we obtain the well-known *h*-index. We prove several properties of the Hirsch function and characterize the shape of general functions that are Hirsch functions. We, moreover, present a formula that enables the calculation of f, given its Hirsch function h_{f}.

**Keywords:**Array, H-function, Hirsch function

## INTRODUCTION

Hirsch^{[}^{1}^{]} introduced the well-known discrete h-index. This indicator was later followed by the *g*-index,^{[}^{2}^{]} and many other variants. The idea of considering (discrete) *h*– and *g*-indices with a variable parameter, originates from van Eck and Waltman.^{[}^{3}^{]} Recently, Lathabai[4] introduced the ψ-index as the indicator with the largest offset-ability. These indicators are discrete, which makes them easy to apply, but it is well-known that for theoretical investigations a continuous version is more feasible. Hence, from now on we will work in a continuous context.

Let f:*R*+→*R* + be a function. Then we define for all θ ϵ*R*_{0}^{+} = *R*^{+} {0}:

We only consider those cases for which (1) has a unique solution. If f(0) = 0 then we exclude a possible extra solution of x = h_{f}(9) = 0 unless this is a unique solution. Figure 1 illustrates some special cases.

Case a). does not lead to a valid solution of (1) as y = 9x and f(x) intersect in more than one point.

Case b). Here f(0) = f’(0) = 0. Here we do not consider x =0, so that (1) has a unique solution for all 9 θ*R*_{0}^{+}.

Case c). Here f(0) = 0 and f’(0) = θ_{0} > 0. We do not consider x = 0 as a solution of (1) if θ > θ_{0} and do consider x = 0 as a solution if 0 < θ ≤ θ_{0}.

Case d). Here we have x = h_{f}(θ) = 0, for all θ ϵ*R*_{0}^{+}.

Although it is possible to solve such special cases differently, the main point is that we know unambiguously what we mean by the notation h_{f}(θ). As h_{f}(θ) is now clearly defined we obtain a well-defined function h_{f}.

**Definition: The Hirsch function**

The function h_{f}: θ ϵ *R*_{0}^{+} *÷* h_{f}(θ) ϵ*R*^{+} is called the Hirsch function.

For θ =1, we obtain the well-known h-index^{[}^{1}^{]} of the continuous function f, explaining the naming of this function. We further note that h_{f} is not defined in point zero so we can say that for f = 0 (the null function) h_{f}(θ) = 0.

The Hirsch function has been used implicitly by Egghe and Rousseau^{[}^{5}^{]} (without naming it as such) and later by Egghe^{[}^{6}^{,}^{7}^{]} while, as mentioned above, the idea of considering h-indices with a variable parameter, originates from van Eck and Waltman.^{[}^{3}^{]}

The Hirsch function is not defined as an explicit function but implicitly through equation (1). We first provide a characterization of such functions.

**Theorem 1**

Let 9 be a function defined on *R*^{+}, continuous in 0. Let further f be a function, continuous in the point φ(0) then the following two statements are equivalent:

Proof. (i) ⇒ (ii)

From (i) and (1) we obtain (2) V9 ∈ *R0+.* For 9 = 0, we find, using the assumed continuity:

Where we have used that we already know (2) for θ > 0. Hence, f(φ(0)) = 0 = 0.φ(0), which is (2) for θ = 0.

(ii) ⇒ (i)

From (2), (1), and the assumed uniqueness we have that Vθϵ *R*_{0}^{+} : h_{f}(θ) = φ(0), by the definition of h_{f} □

Next, we will study the following problems.

(a). Given f, determine h_{f}. This is the formalism shown in (1). We give one simple example: let f(x) = C > 0 (C fixed). Then (1) leads to the equation C = θx. Hence Ɐ *R*_{0}^{+} h_{f}(θ) = x = C/θ. We come to the same result using (2). Indeed, Ɐθϵ *R 0** ^{+}* h

_{f}(θ) = φ(θ) = C/θ.

(b). Given φ, determine f such that φ = h_{f}. This problem already places some extra requirements on φ without which φ = h_{f} is impossible. Consider e.g., the example above: with Ɐ*x∈R*_{0}^{+} φ(x) = C/x. Then (2) leads to . As the range of C/x is *R*_{0}^{+}, f(x) = C on *R*_{0}^{+}and thus also f(x) = C on *R*^{+}by the continuity of f.

(c). Neither f nor φ is given, but a general relationship between f and φ. Here we consider two subcases.

**φ is given via a relation with f**

Example 1. 9 = f (the simplest possible relation). Using (2) we have Ɐθ∈*R*^{+}: f(f(θ)) = 0.f(0). If f is continuous then this relation can only occur if f =0(the null function) or f(x) = x^{α}, where a is the golden section, .[6] Its proof uses the Fibonacci sequence.

Example 2. φ = f_{º}f. Using (2) this leads to f(f(f(θ))) = 0.f(f(θ)), Vθ∈*R*^{+},see Egghe.^{[}^{8}^{]} Again, for f continuous, this requirement leads to two possible solutions, namely f = 0 or f(x) = x^{β}, with β ~ 1.4648493 (smaller than the golden section). Its proof uses a variant of the Fibonacci sequence.

In the same vein, one can consider the case 9=*f* °*f* °…°*f*_{—}*n* times.

**f is given via a relationship with φ**

Example 1. The function f = φ. Although this is essentially the same as the previous example 1, (2) leads to Vθ∈*R*^{+}: φ(φ(θ)) = θ.φ(θ), leading to φ (=f) = 0 or φ(x) (=f(x)) = x^{α}.

Example 2. f = φ^{o}φ

This example is different. Via (2) we find: φ (φ (φ (θ))) = θ.φ(φ(θ)), Ɐθ∈*R*^{+}, see Egghe.^{[}^{6}^{]} For φ continuous, this leads to φ = 0 or φ(x) = x^{α}, with α ~ 1.3247178. hence f(x) = x^{(α2)}.

This ends the introduction. Next, we will study the basic properties of the Hirsch function.

### Properties of the Hirsch function

**Theorem 2**

The function h_{f} is injective on the set {θ ∈*R*_{0}^{+}|| h_{f}(θ) ≠ 0}.

Notation. We denote {θ ∈*R*_{0}^{+}|| h_{f}(θ) ≠ 0} as {h_{f} ≠ 0}.

Proof. Let x_{1} = h_{f}(θ_{1}) = h_{f}(θ_{2}) = x_{2}. Then (1) implies that f(x_{1}) = θ_{1}x_{1} and f(x_{2}) = θ_{2}x_{2}. As x_{1} = x_{2} and f is a function this implies that θ_{1}x_{1} = θ_{2}x_{2}, leading to θ1 = θ2 if x1 = x2 ≠ θ. □

The next theorem provides a new characterization of h_{f}.

**Theorem 3**

Let m be a function of functions m: f → m(f), then the following statements are equivalent:

(i) m(f) = h_{f}

(ii) V 0 ∈ *R*_{0}^{+} : mθ(f) = , where ψ_{f} is injective, and defined as:

Proof. (i) ⇒ (ii)

From (i) and (1) it follows that Ɐ θ∈ *R*_{0}^{+}: x = m_{θ}(f) ⇔ f(x) = θx ⇔ θ = f(x)/x ⇔ x = 9_{f}^{-1}(*θ*). Moreover, from the fact that h_{f} is a function, it follows that ψ_{f} is injective.

(ii) ⇒ (i)

It follows similarly from (ii) and (1) that m(f) = h_{f}.

### Remark

As *h*_{f}^{-1}*= ψ*_{f}*,* with *ψ** _{f}* defined in (3) it follows that

*h*

_{f}

^{–}^{1}is a function on

*R*

_{0}

^{+}. This immediately leads to (see also Theorem 2):

The two implications in (4) do not hold for h_{f} (see further). To study this, we recall two results (stated as lemmas) from real analysis.

#### Lemma 1

If f is continuous on an interval (possibly infinitely long) and injective then f is strictly monotonous.

#### Lemma 2

If f is injective, then the following two statements are equivalent:

(i) f is continuous on [a,b]

(ii) The function f^{-1} is continuous on [f(a), f(b)] (or [f(b), f(a)])

The proof can be found using Lemma 1 and,^{[}^{9}^{]} Theorem 2.27.

Notation. The domain of a function f is denoted as D(f).

#### Theorem 4

If D(f) is an interval, then f is continuous implies that h_{f} is continuous.

Proof. D(*ψ** _{f}*) = D(f) {0}, hence an interval. If f is continuous then also

*ψ*

*is continuous and*

_{f}*ψ*

*is an injection (by Theorem 3). Applying now Lemma 2 onψ*

_{f}_{f}shows that

*ψf*

^{-1}is a continuous function. It then follows from Theorem 3 that h

_{f}=

*ψ*

_{f}^{-1}is also continuous. □

Theorem 4 does not hold if one removes the requirement that D(f) is an interval. This is illustrated in Figure 2.

We know that f is continuous if and only if is continuous. Yet, we will show that the implication h_{f} continuous ^ f continuous does not always hold. For this, we need some preliminary results.

Lemma 3. If f:*R*^{+}→*R*^{+} is continuous and injective on {f^0}, then one of the following three statements hold:

(i) f is injective

(ii) ∃y_{0}>0 such that and f (> 0) is injective, hence strictly increasing, on] y_{0}, + ∞,

(iii) ∃*x*_{0}≥0 such that and f (> 0) is injective, hence strictly decreasing, on [0, x_{0} [. Note that if x_{0} = 0, this includes the case f = 0.

Proof. Assume (i) is not the case, i.e., f is not injective. Yet, we know that f is injective on {f ^ 0}. Hence, there exist x, y, 0 ≤ x < y such that f(x) = f(y) = 0.

Indeed, otherwise, there would exist z ∈ ]x,y[ such that f(z) ≠ 0. Because f is continuous it assumes all values between f(x) = 0 and f(z) > 0 on ]x,z[ and similarly on the interval ]z,y[. Consequently, there exist points x’ and y’, x’ ∈ ]x,z[ and y’ ∈ ]z,y[, (hence x’ ^ y’) such that f(x’) = f(y’) = f(z)/2 ≠ 0, which contradicts the fact that f is injective on {f ^ 0}.

Next, we show that

Assume this is not the case. Then there exists u ∈ [0,x[ such that f(u) > 0 and v ∈ ]y, +∞[ such that f(v) > 0. As f is continuous it takes all values between f(x) = 0 en f(u) > 0 on ]u,x[ and between f(y) = 0 and f(v) > 0 on ]y,v[. Put a = min(f(u), f(v)) > 0. Then there exist x’ in ]u,x[ and y’ in ]y,v[ such that f(y’) = f(x’) = a ^ 0 (and x’ ^ y’). This is in contradiction with the fact that f is injective on {f ≠ 0}.

From (*) and (**) it follows that f|_{[0,y]} = 0 or f|_{[x, +o[} = 0, with 0 ≤ x < y. In the first case we set y_{0} = sup {y > 0 such that f|_{[0y]} = 0}. Then we know that and not equal to zero (hence strictly positive) on the complement of [0,y_{0}]. From this, it follows that f is injective. Then f is strictly increasing on] y_{0}, +o [and (ii) has been proved.

In the second case we set x_{0} = inf {x ≥ 0 such that f|_{[x,+∞[} = 0}. Then and on the complement of [x_{0}, +∞> [f≠ 0 and hence injective. In this case, f decreases strictly on [0,x_{0}[ and (ii) is proved. Y

### Corollary

A continuous Hirsch function h_{f} on *R*_{0}^{+} is of one of the following three types:

(i) h_{f} is injective on *R*_{0}^{+};

(ii) h_{f} = 0 on an interval [0,y_{0}], y_{0} > 0 and strictly increasing on the compliment;

(iii) h_{f} = 0 on an interval [x_{0}, + o [, x_{0} > 0 and strictly decreasing on the complement; including the case h_{f} = 0.

Proof. This follows from Lemma 3 and Theorem 2, with f (in Lemma 3) replaced by h_{f}, defined on *R*_{0}^{+}.Y

We show that these three types occur.

(i) This class is best known as it includes the functions f(x) = x^{c}, c > 1. Then, Ɐθ ∈R_{0}^{+} : , which is a strictly increasing injection. Note that for 1/(c-1) = c we find c = α and hence f = hf on R_{0}^{+}.

(ii). See Figure 3a. The function f is strictly convex, θ_{0} = f’(0) > 0. By definition, h_{f} is zero on [0,θ_{0}] and strictly increasing on [ *θ*_{0},+∞].

The consequence of Lemma 3 also provides conditions for an equation such as (2), Theorem 1, to have or not to have a solution,

These cases are discussed in the next theorem.

#### Theorem 5

If 9 is a continuous function *R*^{+}→ *R*^{+} which is not of the form (i), (ii), or (iii) of the above corollary then a function f such that h_{f} = φ does not exist. If φ is of the form (i), (ii), or (iii) then the solution of (2), namely h_{f} = φ is given by

with φ^{-1} the inverse function of the injective part of φ (abuse of notation). This function φ^{-1} always exists, except when x_{0} = 0 in (iii), in which case φ = 0 and φ = h_{f} with f = 0. We further note that in cases (ii) and (iii) f(0) = 0.

#### Proof

Case (i). In this case, φ is injective and (2) gives:

Denoting φ(θ) by x we find that φ^{-1}(x) = 0, which yields (5).

Case (ii). Now we know that there exists y_{0} > 0 such that and φ is strictly increasing (hence injective) on ]y_{0}, + ∞[. Next‚weset f(x) = *x*φ^{-1}(*x*) on φ([y_{0},+∞[). As φ is continuous and In this way, f is defined on *R*_{0}^{+} with (2) holding on [y_{0}, + ∞]. Now define f(0) = 0, then we have, Ɐ *θ* ∈[0,y_{0}]:

showing that (2) holds on *R*^{+} and thus, by Theorem 1, h_{f} = φ on *R*_{0}^{+}.

Case (iii) Now we know that there exists x_{0} ≥ 0 such that with φ strictly decreasing (and hence injective) on [0,x_{0}]. For x_{0} = **0**, φ = 0 on *R*^{+} and we take f = **0** on *R*^{+}, leading to h_{f} = φ on *R*_{0}^{+} (by (1)).

Assume now that x_{0} > 0. Define f(x) = x φ^{-1}(x) on φ([0,x_{0}[)≠∅. As φ is continuous and . So far, we defined f on *R*_{0}^{+}, with (2) holding on [0, x_{0}[. Now, put f(0) = 0, then we have Ɐθ∈[*x*_{0},+0[:

showing again that (2) holds on *R*^{+} and thus, by Theorem 1, h_{f} = φ on *R*0^{+}. □

### Practical conclusion

Leaving x = 0 aside we see that the solution of h_{f} = φ is given by equation (5) with φ^{-1} the inverse of φ on the injective part of φ (and f = 0 for φ = 0).

#### Examples

(i) For φ(x) = C/x, C>0 constant, we see that φ is injective and φ^{-1} = φ. Then (5) yields: f(x) = x.C/x = C and h_{f} = φ.

For φ(x) = x^{c}, φ^{-1} (x) = x^{1/}^{c} and, by (5), ; h_{f} = φ.

(ii) and (iii). These cases are similar so we give just one example.

For we see that 9 is strictly increasing on [b, + ∞], and hence injective. On this set the function φ^{-1} (x) = b + log_{a}(x+1) and hence, using (5) we have:

Ɐ*x*>0; f(x) = x (b + log_{a}(x+1)) and f(0) = 0, showing that h_{f} = φ. Note that f’(0) = b and that h_{f} is zero on [0, b].

Finally, we come to the case “h_{f} continuous implies f continuous”, the inverse statement of Theorem 4.

#### Theorem 6

If the range of f, denoted as R(f) is an interval, then h_{f} continuous implies f continuous on D(f) ∩ *R*_{0}^{+}.

#### Proof

As h_{f} is continuous everywhere, it is also continuous on {h_{f} ≠ 0}, which is an interval inside , by the corollary to Lemma 3. By Theorem 3 , which too is an interval because R(f) is an interval. By Theorem 2 we know that h_{f} is injective on {h_{f} ≠ 0}. Then it follows from Lemma 2 that h_{f} ^{-1} = ψ*f* (by Theorem 3) is continuous on . Finally, as f(x) = x *ψ** _{f}*(

*x*) on

*R*

_{0}

^{+}(by Theorem 3), this shows that f is continuous on D(f) ∩

*R*

_{0}+.Y

The next example shows that f is not necessarily continuous in zero. Take

Then R(f) = *R*_{0}^{+}, which is an interval, h_{f} is continuous on *R*_{0}^{+} but f is not continuous in 0, see Figure 4.

We finish this article by remarking that the condition “R(f) is an interval” is necessary for Theorem 6. Figure 5 provides an example of a function f which is not continuous on *R* _{0}^{+}, but h_{f} is continuous because D(h_{f}) is not an interval, (because R(f) is not an interval).

## CONCLUSION

This article illustrates how a very practical tool, here the h-index, can inspire further developments, enriching theoretical informetric. The originality of this contribution lies in the finding that – essentially – the *h*-function is injective and that the original function can be recovered from the *h*-function by multiplying the inverse of this function with the independent variable x. Our results show the simplicity of the *h*-index tool.

Returning to the introduction, it seems natural to try to construct a similar theory based on the *g*-index or the *V* – index. Yet we have not been able to realize this, mainly because sums (or integrals) occur in these indices, while this is not the case for the h-index. This makes studying inverses (which we need), not a sinecure. It can be stated though that a theory like we developed for the *h*-index is possible for other indices, such as Kosmulski’s.^{[}^{10}^{]} We did not include this because it brings nothing new and is just more difficult.

The author hopes that his theoretical study will inspire colleagues to further useful developments or interesting applications.

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