Stationary states in gas networks

1.   Introduction and definitions

Let $ \mathfrak{A} $ denote the family of all functions $ f $ which are analytic in the open unit disc $ \mathcal{U} = \left \{ z:\left \vert z\right \vert < 1\right \} $ and satisfying the normalization

while by $ \mathcal{S} $ we mean the class of all functions in $ \mathfrak{A} $ which are univalent in $ \mathcal{U}. $ Also let $ \mathcal{S}^{\ast } $ and $ \mathcal{C} $ denote the familiar classes of starlike and convex functions, respectively. If $ f $ and $ g $ are analytic functions in $ \mathcal{U }, $ then we say that $ f $ is subordinate to $ g, $ denoted by $ f\prec g, $ if there exists an analytic Schwarz function $ w $ in $ \mathcal{U} $ with $ w\left(0\right) = 0 $ and $ \left \vert w(z)\right \vert < 1 $ such that $ f(z) = g\left(w(z)\right). $ Moreover if the function $ g $ is univalent in $ \mathcal{U} $, then

For arbitrary fixed numbers $ A, $ $ B $ and $ b $ such that $ A $, $ B $ are real with $ -1\leq B < A\leq 1 $ and $ b\in \mathbb{C} \backslash \{0\} $, let $ \mathcal{P}[b, A, B] $ denote the family of functions

analytic in $ \mathcal{U} $ such that

Then, $ p\in \mathcal{P}[b, A, B] $ can be written in terms of the Schwarz function $ w $ by

By taking $ b = 1-\sigma $ with $ 0\leq \sigma < 1, $ the class $ \mathcal{P} [b, A, B] $ coincides with $ \mathcal{P}[\sigma, A, B], $ defined by Polatoğ lu ^[17,18] (see also ^[2]) and if we take $ b = 1, $ then $ \mathcal{ P}[b, A, B] $ reduces to the familiar class $ \mathcal{P}[A, B] $ defined by Janowski ^[10]. Also by taking $ A = 1, $ $ B = -1 $ and $ b = 1 $ in $ \mathcal{P} [b, A, B] $, we get the most valuable and familiar set $ \mathcal{P} $ of functions having positive real part. Let $ \mathcal{S}^{\ast }[A, B, b] $ denote the class of univalent functions $ g $ of the form

in $ \mathcal{U} $ such that

Then $ S^* [A, B] : = S^* [A, B, 1] $ and the subclass $ \mathcal{S}^{\ast}[1,-1,1] $ coincides with the usual class of starlike functions.

The set of Bazilevič functions in $ \mathcal{U} $ was first introduced by Bazilevič ^[7] in 1955. He defined the Bazilevič function by the relation

where $ p\in \mathcal{P}, $ $ g\in \mathcal{S}^{\ast }, $ $ \beta $ is real and $ \alpha > 0 $. In 1979, Campbell and Pearce ^[8] generalized the Bazilevi č functions by means of the differential equation

where $ \alpha +i\beta \in \mathbb{C} -\left \{ \text{negative integers}\right \}. $ They associate each generalized Bazilevič functions with the quadruple $ (\alpha, \beta, g, p). $

Now we define the following subclass.

Definition 1.1. Let $ g $ be in the class $ \mathcal{S}^{\ast }[A, B] $ and let $ p\in \mathcal{P} [b, A, B] $. Then a function $ f $ of the form $ \left(1.1 \right) $ is said to belong to the class of generalized Bazilevič function associated with the quadruple $ (\alpha, \beta, g, p) $ if $ f $ satisfies the differential equation

where $ \alpha + i\beta \in \mathbb{C} - \{{\text {negative integers}} \} $.

The above differential equation can equivalently be written as

or

Since $ p\in \mathcal{P}[b, A, B], $ it follows that

where $ g\in \mathcal{S}^{\ast }[A, B]. $

Several research papers have appeared recently on classes related to the Janowski functions, Bazilevič functions and their generalizations, see ^{[3,4,5,13,16,21,22]}.

2.   Lemmas

The following are some results that would be useful in proving the main results.

Lemma 2.1. Let $ p\in \mathcal{P}[b, A, B] $ with $ b\neq 0, $ $ -1\leq B < A\leq 1, $ and has the form $ \left(1.2\right). $ Then for $ z = re^{i\theta }, $

Proof. The proof of this lemma is straightforward but we include it for the sake of completeness. Since $ p\in \mathcal{P}[b, A, B] $, we have

Let $ \tilde{p}(z) = 1+\sum_{n = 1}^{\infty }c_{n}\, z^{n} $. Then

Comparing the coefficients of $ z^{n} $, we have

Since $ \left \vert c_{n}\right \vert \leq A-B $ ^[20], it follows that $ \left \vert p_{n}\right \vert \leq \left \vert b\right \vert (A-B) $ and so

Thus the proof is complete.

Lemma 2.2. ^[1] Let $ \Omega $ be the family of analytic functions $ \omega $ on $ \mathcal{U} $, normalized by $ \omega (0) = 0 $, satisfying the condition $ \left \vert \omega (z)\right \vert < 1 $. If $ \omega \in \Omega $ and

then for any complex number $ t $,

The above inequality is sharp for $ \omega (z) = z $ or $ \omega (z) = z^{2} $.

Lemma 2.3. Let $ p(z) = 1+\sum \limits_{n = 1}^{\infty }p_{n}z^{n}\in \mathcal{P}[b, A, B], $ $ b\in \mathbb{C}\backslash \{0\}, $ $ -1\leq B < A\leq 1. $ Then for any complex number $ \mu $,

This result is sharp.

Proof. Let $ p\in \mathcal{P}[b, A, B] $. Then we have

or, equivalently

which would further give

Comparing the coefficients of $ z $ and $ z^{2} $, we obtain

By a simple computation,

Now by using Lemma 2.2 with $ t = \mu b(A-B)+B $, we get the required result. Equality holds for the functions

Now we prove the following result by using a method similar to the one in Libera ^[12].

Lemma 2.4. Suppose that $ N $ and $ D $ are analytic in $ \mathcal{U} $ with $ N(0) = D(0) = 0 $ and $ D $ maps $ \mathcal{U} $ onto a many sheeted region which is starlike with respect to the origin. If $ \frac{N^{\prime }(z)}{D^{\prime }(z) }\in \mathcal{P}[b, A, B] $, then

Proof. Let $ \frac{N^{\prime }(z)}{D^{\prime }(z)}\in \mathcal{P}[b, A, B]. $ Then by using a result due to Attiya ^[6], we have

where $ c(r) = \frac{1-B\left[B+b\left(A-B\right) \right] r^{2}}{1-B^{2}r^{2}} $ and $ d\left(r\right) = \frac{\left \vert b\right \vert \left(A-B\right) r^{2}}{1-B^{2}r^{2}}. $ We choose $ A\left(z\right) $ such that $ \left \vert A\left(z\right) \right \vert < d\left(r\right) $ and

Now for a fixed $ z_{0} $ in $ \mathcal{U} $, consider the line segment $ L $ joining $ 0 $ and $ D\left(z_{0}\right) $ which remains in one sheet of the starlike image of $ \mathcal{U} $ by $ D. $ Suppose that $ L^{-1} $ is the pre-image of $ L $ under $ D $. Then

This implies that

Therefore

For $ A = -B = b = 1 $, we have the following result due to Libera ^[12].

Lemma 2.5. If $ N $ and $ D $ are analytic in $ \mathcal{U} $ with $ N(0) = D(0) = 0 $ and $ D $ maps $ \mathcal{U} $ onto a many sheeted region which is starlike with respect to the origin, then

Lemma 2.6. ^[14] If $ -1\leq B < A\leq 1, \beta _{1} > 0 $ and the complex number $ \gamma $ satisfies $ {Re}\left \{ \gamma \right \} \geq -\beta_{1}\left(1-A\right) /\left(1-B\right), $ then the differential equation

has a univalent solution in $ \mathcal{U} $ given by

If $ p\left(z\right) = 1+p_{1}z+p_{2}z^{2}+\cdots $ is analytic in $ \mathcal{U} $ and satisfies

then

and $ q\left(z\right) $ is the best dominant.

3.   Some auxiliary results

Before proving the results for the generalized Bazilevič functions, let us discuss a few results related to the function $ g \in S^*[A, B] $.

Theorem 3.1. Let $ g\in \mathcal{S}^{\ast }[A, B] $ and of the form $ \left(1.3\right) $. Then for any complex number $ \mu, $

Proof. The proof of the result is the same as of Lemma 2.3. The result is sharp and equality holds for the function defined by

or

Theorem 3.2. Let $ g\in \mathcal{S}^{\ast }[A, B]. $ Then for $ c > 0, $ $ \alpha > 0 $ and $ \beta $ any real number,

is in $ \mathcal{S}^{\ast }[A, B] $. In addition

where

Proof. From $ \left(3.1\right), $ we have

Differentiating and rearranging gives

where $ p(z) = \frac{zG^{\prime }(z)}{G(z)} $. Then differentiating $ \left(3.2\right) $ logarithmically, we have

Since $ g\in \mathcal{S}^{\ast }[A, B] $, it follows that

Now by using Lemma 2.6, for $ \beta _{1} = \alpha $ and $ \gamma = c+i\beta, $ we obtain

where

Now by using the properties of the familiar hypergeometric functions proved in ^[15], we have

This implies that

and

Theorem 3.3. Let $ g\in \mathcal{S}^{\ast }[A, B]. $ Then

is $ (\alpha +c) $-valent starlike, where $ \alpha > 0, $ $ c > 0 $ and $ \beta $ is a real number.

Proof. Let $ D_{1}(z) = zS^{\prime }(z) = z^{c+i\beta }g^{\alpha }(z) $ and $ N_{1}(z) = S(z). $ Then

Since $ g\in \mathcal{S}^{\ast }[A, B]\subset \mathcal{S}^{\ast }\left(\frac{ 1-A}{1-B}\right) $, see ^[10], it follows that

Also

Now by using Lemma 2.5$, $ we have

By the mean value theorem for harmonic functions,

Therefore

Now by using a result due to [9,p 212], we have that $ S $ is $ (c+\alpha) $-valent starlike function.

4.   Main results

Now we are ready to discuss some results related to the defined generalized Bazilevič functions.

Theorem 4.1. Let $ f $ be a generalized Bazilevič function associated by the quadruple $ \left(\alpha, \beta, g, p\right), $ where $ g\in \mathcal{S} ^{\ast }[A, B] $ of the form $ \left(1.3\right) $ and $ p\in \mathcal{P} [b, A, B] $ of the form $ \left(1.2\right) $. Then for $ c > 0, $

is a generalized Bazilevič function associated by the quadruple $ (\alpha, \beta, G, p), $ where $ G\in \mathcal{S}^{\ast }[A, B, \delta] $, as defined by $ \left(3.1\right). $

Proof. From $ \left(4.1\right) $, we have

This implies that

Differentiate both sides and rearrange, we get

and

Now from $ (3.1) $, we have

With this, note that

which implies $ \frac{N^{\prime }(z)}{D^{\prime }(z)} \in \mathcal{P}[b, A, B] $. By Theorem 3.3, we know that $ D(z) = \int_{0}^{z}t^{c+i\beta -1}g^{\alpha }(t)dt $ is $ (\alpha +c) $-valent starlike. Therefore by using Lemma 2.4, we obtain

This is the equivalent form of Definition 1.1. Hence the result follows.

Corollary 4.2. Let $ A = 1, B = -1 $ and $ \beta = 0 $ in Theorem 4.1. Then

belong to $ S^{\ast }\left(\delta _{1}\right) $, where

Hence $ G $ is starlike when $ g\in \mathcal{S}^{\ast }, $ and

belongs to the class of Bazilevič functions associated by the quadruple $ (\alpha, 0, G, p). $

Theorem 4.3. Let $ f $ of the form $ \left(1.1\right) $ be a generalized Bazilevič function associated by the quadruple $ (\alpha, \beta, g, p) $, with $ g\in \mathcal{S}^{\ast }[A, B] $ of the form $ \left(1.3\right) $ and $ p\in \mathcal{P}[b, A, B] $ of the form $ \left(1.2\right) $. Then

This inequality is sharp.

Proof. Since $ f $ is a generalized Bazilevič function associated by the quadruple $ (\alpha, \beta, g, p) $, we have

As $ f, \ g $ and $ p $ respectively have the form $ \left(1.1\right), \left(1.3\right) $ and $ \left(1.2\right), $ it is easy to get

Putting these values in $ \left(4.2\right) $ and comparing the coefficients of $ z, $ we obtain

Similarly by comparing the coefficients of $ z^{2} $ and rearranging, we have

From $ \left(4.4\right) $, we have

Now by using Theorem 3.1 and Lemma 2.3, both with $ \mu = \frac{1}{2} $, we obtain

and

Therefore, we have

The equality

for $ B\neq 0 $ can be obtained for

Similarly, the equality

for $ B = 0 $ can be obtained for the function $ g_{\ast }(z) = $ $ ze^{\frac{A}{2} z^{2}} = z+\frac{A}{2}z^{3}+\cdots. $ Also equality for the functional $ \left \vert p_{2}-\frac{1}{2}p_{1}^{2}\right \vert $ can be obtained by the functions

Corollary 4.4. For $ A = 1, $ $ B = -1 $ and $ b = 1 $, we have the result proved in ^[8]:

For $ \alpha = 1 $, $ \beta = 0 $, we have $ f\in \mathcal{K} $, the class of close-to-convex functions, and

The latter result has been proved in ^[11].

Theorem 4.5. Let $ f $ of the form $ \left(1.1\right) $ be a generalized Bazilevič function associated by the quadruple $ (\alpha, \beta, g, p) $, with $ g\in \mathcal{S}^{\ast }[A, B] $ and of the form $ \left(1.3\right) $ and $ p\in \mathcal{P}[b, A, B] $ of the form $ \left(1.2\right) $. Then

(i)

(ii) If $ \alpha = 0, $ then

Both the above inequalities are sharp.

Proof. (ⅰ) From $ \left(4.3\right) $, we have

This implies that

By using the coefficient bound for $ \mathcal{S}^{\ast }[A, B] $ along with the coefficient bound of $ \mathcal{P}[b, A, B] $, we have

This implies that

Equality can be obtained by the functions

(ⅱ) Let $ \alpha = 0. $ Then from $ \left(4.3\right) $ and $ \left(4.4 \right) $, we have

This implies

where $ \mu = \frac{1}{2}-\frac{(3+i\beta)}{2\left(1+i\beta \right) ^{2}}. $ Now by using Lemma 2.3, we obtain

Sharpness can be attained by the functions

Corollary 4.6. For $ A = 1, $ $ B = -1 $ and $ b = 1 $, we have

and

In the final part of this paper, we look at some results for the generalized Bazilevič functions associated with $ \beta = 0 $.

Let $ C_{r} $ denote the closed curve which is the image of the circle $ \left \vert z\right \vert = r < 1 $ under the mapping $ w = $ $ f(z) $, and $ L_{r}\left(f(z)\right) $ denote the length of $ C_{r} $. Also let $ M(r) = {\max} _{\left \vert z\right \vert = r} \left \vert f(z)\right \vert \ $and $ m(r) = { \min}_{\left \vert z\right \vert = r} \left \vert f(z)\right \vert $. We now prove the following result.

Theorem 4.7. Let $ f $ be a generalized Bazilevič function associated by the quadruple $ (\alpha, 0, g, p) $. Then for $ B\neq 0 $,

where

Proof. As $ f $ is a generalized Bazilevič function associated by the quadruple $ (\alpha, 0, g, p) $, we have

where $ g\in \mathcal{S}^{\ast }[A, B] $ and $ p\in \mathcal{P}[b, A, B] $. Since for $ z = re^{i\theta } $, $ 0 < r < 1 $,

we have for $ 0 < \alpha \leq 1 $,

where $ \frac{zg^{\prime }(z)}{g(z)} = h\left(z\right) \in \mathcal{P}[A, B] $. Now by using the distortion theorem for Janowski starlike functions when $ B\neq 0 $ (see ^[10]) and the Cauchy-Schwarz inequality, we have

Now by using Lemma 2.1 for both the classes $ \mathcal{P}[b, A, B] $ and $ \mathcal{P}[A, B] $, along with the result

for $ p\in \mathcal{P}[b, A, B] $ (see ^[19]), we can write

Since $ 1-\left \vert B\right \vert r\geq 1-r $ and $ 1- B^2 r^2\geq 1-r^2 $,

where $ C(\alpha, b, A, B) = 2\pi |b| [(A-B) + (1/ \alpha)] B $.

When $ \alpha > 1, $ we can prove similarly as above to get

Corollary 4.8. For $ g\in \mathcal{S}^{\ast } $ and $ p\in \mathcal{P}(b), $ we have

Theorem 4.9. Let $ f $ be a generalized Bazilevič function associated by the quadruple $ (\alpha, 0, g, p) $, where $ g\in \mathcal{S}^{\ast }[A, B] $ and $ p\in \mathcal{P} [b, A, B] $. Then for $ B\neq 0 $,

Proof. By Cauchy's theorem for $ z = re^{i\theta }, $ $ n\geq 2, $ we have

Therefore

By using Theorem 4.7 for the case $ 0 < \alpha \leq 1, $ we have

Hence, by taking $ r $ approaches $ 1^{-} $,

For $ \alpha > 1, $ we have

Theorem 4.10. Let $ f $ be a generalized Bazilevič function represented by the quadruple $ (\alpha, 0, g, p), $ where $ g\in \mathcal{S}^{\ast }[A, B] $ and $ p\in \mathcal{P }[b, A, B]. $ Then for $ B\neq 0 $,

Proof. Since $ f $ is a generalized Bazilevič function associated by the quadruple $ (\alpha, 0, g, p), $ by definition, we have

where $ g\in \mathcal{S}^{\ast }[A, B] $ and $ p\in \mathcal{P}[b, A, B]. $ This implies that

and so

Now by using the results

and

we have

Putting $ s = ru, $ we have

where $ {_{2}}F_{1} \left(a; b;c; z\right) $ is the hypergeometric function.

Corollary 4.11. For $ g\in \mathcal{S}^{\ast \mathit{\text{}}} $and $ p\in \mathcal{P}, $ we have

Acknowledgments

The research for the fourth author is supported by the USM Research University Individual Grant (RUI) 1001/PMATHS/8011038.

Conflict of interest

The authors declare that they have no conflict of interests.