Estimates of Initial Coefficients for Bi-Univalent Functions

We consider the Fekete-Szegö inequalities for classes which were defined by Murugusundaramoorthy et al. (2013). These inequalities will result in bounds of the third coefficient which are better than these obtained by Murugusundaramoorthy et al. (2013). Moreover, we discuss two other classes of bi-univalent functions. The estimates of initial coefficients in these classes are obtained

Second Hankel Determinants for the Class of Typically Real Functions

We discuss the Hankel determinants H2(n)=anan+2-an+12 for typically real functions, that is, analytic functions which satisfy the condition Im ⁡z Im⁡f(z)≥0 in the unit disk Δ. Main results are concerned with H2(2) and H2(3). The sharp upper and lower bounds are given. In general case, for n≥4, the results are not sharp. Moreover, we present some remarks connected with typically real odd functions

On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis

In this paper we consider two functionals of the Fekete–Szegö type: $\Phi _f(\mu ) = a_2 a_4-\mu a_3{}^2$ and $\Theta _f(\mu ) = a_4-\mu a_2a_3$ for analytic functions $f(z) = z+a_2z^2+a_3z^3+\ldots$, $z\in \Delta$, ($\Delta = \lbrace z\in \mathbb{C}:|z|<1\rbrace$) and for real numbers $\mu$. For $f$ which is univalent and convex in the direction of the imaginary axis, we find sharp bounds of the functionals $\Phi _f(\mu )$ and $\Theta _f(\mu )$. It is possible to transfer the results onto the class $\mathcal{K}_{\mathbb{R}}(i)$ of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class $\mathcal{T}$ of typically real functions. As corollaries, we obtain bounds of the second Hankel determinant in $\mathcal{K}_{\mathbb{R}}(i)$ and $\mathcal{T}$

On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis

In this paper we consider two functionals of the Fekete–Szegö type: $\Phi _f(\mu ) = a_2 a_4-\mu a_3{}^2$ and $\Theta _f(\mu ) = a_4-\mu a_2a_3$ for analytic functions $f(z) = z+a_2z^2+a_3z^3+\ldots$, $z\in \Delta$, ($\Delta = \lbrace z\in \mathbb{C}:|z|<1\rbrace$) and for real numbers $\mu$. For $f$ which is univalent and convex in the direction of the imaginary axis, we find sharp bounds of the functionals $\Phi _f(\mu )$ and $\Theta _f(\mu )$. It is possible to transfer the results onto the class $\mathcal{K}_{\mathbb{R}}(i)$ of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class $\mathcal{T}$ of typically real functions. As corollaries, we obtain bounds of the second Hankel determinant in $\mathcal{K}_{\mathbb{R}}(i)$ and $\mathcal{T}$

Sharp bounds of the third Hankel determinant for classes of univalent functions with bounded turning

summary:We improve the bounds of the third order Hankel determinant for two classes of univalent functions with bounded turning

Modelowanie przepływu dyspersji cementowej jako cieczy nieliniowo plastycznie lepkiej

The analytical solution for pipeline and annular flow of the cement dispersion for non-linear viscoplastic model of the fluid, compared with another reological models were presented in the paper. The calculations of the pressure gradient and injection range shows that it is necessary to take into account the friction factor in the theoretical analysis of the cement grout flow.W artykule zaprezentowano rozwiązanie analityczne przepływu dyspersji cementowej w prostoliniowych przewodach cylindrycznych, jako cieczy nieliniowo plastycznie lepkiej na tle innych modeli reologicznych. Przeprowadzone obliczenia rozkładu ciśnienia tłocznego oraz zasięgu przepływu wskazują na konieczność uwzględniania oporów natury tarciowej w analizie przepływu mieszanki cementowej

Hankel Determinants for Univalent Functions Related to the Exponential Function

Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f ( 0 ) = f ′ ( 0 ) − 1 = 0 and z f ′ ( z ) f ( z ) ≺ e z . On the other hand, g ∈ K e if and only if z g ′ ∈ S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class