In this article, some Bohr inequalities for analytical functions on the unit disk are generalized to forms with two parameters. One of our results is sharp.

In this paper, several Bohr-type inequalities are generalized to the form with two parameters for bounded analytic function. Most of results sharp.

In this article, Bohr type inequalities for some complex valued harmonic functions defined on the unit disk are given. All results sharp.

In this paper we find the best possible lower power mean bounds for Neuman-Sándor and present sharp ratio of identric means.

In this paper, we find the largest value α and least β such that double inequality L (a,b) < M(a,b) holds for all a,b > 0 with a = b .Here, p are Neuman-Sándor -th generalized logarithmic means of , respectively.

In this paper, we prove three sharp inequalities as follows: , and for all with . Here, are the r th generalized logarithmic, Neuman-Sándor, first second Seiffert means of a b, respectively. MSC:26E60.

For , the generalized logarithmic mean arithmetic and geometric of two positive numbers are defined by for respectively. In this paper, we find greatest value (or least resp.) such that inequality holds all with .

We answer the question: for α , β γ ∈ (0,1) with + = 1, what are greatest value p and least q such that double inequality L ( a b ) &lt; A G H holds all &gt; 0 ≠ ? Here ), denote generalized logarithmic, arithmetic, geometric, harmonic means of two positive numbers respectively.

For p ∈ R the -th power mean M (a,b) , arithmetic A(a,b) geometric G(a,b) and harmonic H(a,b) of two positive numbers a b are defined by, = 2ab/(a + b) respectively.In this paper, we answer questions: α (0,1) what greatest values p, r m least q, s n such that inequalitiesMathematics subject classification (2010): 26E60.

For , the power mean of order two positive numbers and is defined by for . In this paper, we answer question: what are greatest value least such that double inequality holds all with ? Here denote classical arithmetic, geometric, harmonic means, respectively.

Abstract In this article, we answer the question: For p , ω ∈ ℝ with &gt; 0 and ( - 2) ≠ 0, what are greatest value r 1 = ) least 2 such that double inequality <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mfenced> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mfenced> <mml:mo>&lt;</mml:mo> <mml:mi>H</mml:mi> <mml:mi>p</mml:mi> <mml:mi>ω</mml:mi>...

In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions completely monotone sequences.

In this paper, for the convolution and convex combination of harmonic mappings, radii univalence, full convexity starlikeness order $\unicode[STIX]{x1D6FC}$ are explored. All results sharp. By way application, univalent radius Bloch constant two bounded mappings obtained.

In the article, we prove that double inequality $$ \alpha L(a,b)+(1-\alpha)T(a,b)< \mathit{NS}(a,b)< \beta L(a,b)+(1-\beta)T(a,b) holds for $a,b>0$ with $a\ne b $ if and only $\alpha\ge1/4$ $\beta\le1-\pi/[4\log(1+\sqrt{2})]$ , where $\mathit{NS}(a,b)$ $L(a,b)$ $T(a,b)$ denote Neuman-Sándor, logarithmic second Seiffert means of two positive numbers a b, respectively.

A planar harmonic mapping is a complex-valued function f : U ? C of the form (x + iy) = u(x,y) iv(x,y), where u and v are both real harmonic. Such can be written as h g?, g analytic; g'=h' called dilatation f. We consider linear combinations mappings that vertical shears asymmetrical strip j(z) 1/2isin?j log (1+zei?j/ 1+ze-i?j) with various dilatations, ?j [?/2,?), j=1,2. prove sufficient conditions for combination this class univalent to convex in direction imaginary axis.

We present the best possible power mean bounds for product any p &gt; 0, α ∈ (0,1), and all a , b 0 with ≠ . Here, M ( ) is th of two positive numbers

We call the solution of a kind second order homogeneous partial differential equation as real kernel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha minus"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha -</mml:annotation> </mml:semantics> </mml:math> </inline-formula>harmonic mappings. For this class mappings, we explore its Heinz type...

In this paper, we find the least value α and greatest β such that double inequality $$P^{\alpha}(a,b)T^{1-\alpha}(a,b)< M(a,b)< P^{\beta}(a,b)T^{1-\beta}(a,b) $$ holds for all $a,b>0$ with $a\neq b$ , where $M(a,b)$ $P(a,b)$ $T(a,b)$ are Neuman-Sándor, first second Seiffert means of two positive numbers a b, respectively.

Abstract In this article, we establish a double inequality between the generalized Heronian and logarithmic means. The achieved result is inspired by articles of Lin Shi et al., methods from Janous. inequalities obtained improve existing corresponding results and, in some sense, are optimal. 2010 Mathematics Subject Classification: 26E60.

We present sharp upper and lower generalized logarithmic mean bounds for the geometric weighted of harmonic means.

In this paper, we establish several inequalities for the generalized weighted quasi-arithmetic integral mean by use of Chebyshev inequality, Jensen inequality and convexity.

We call a kind of mappings induced by weighted Laplace operator as complex valued kernel $\alpha$-harmonic mappings. In this article, for class mappings, the Heinz type lemma is established, and best inequality obtained. Next, extremal function Schwartz's Lemma discussed. Finally, coefficients are estimated subclass alpha harmonic whose real numbers.