A new subclass of the starlike functions

Motivated by the R{\o}nning--starlike class [Proc Amer Math Soc {\bf118}, no. 1, 189--196, 1993], we introduce new class $\mathcal{S}^*_c$ includes of analytic and normalized functions $f$ which satisfy the inequality\begin{equation*}  {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\}\geq\left|\frac{f(z)}{z}-1\right|\quad(|z|\end{equation*}In this paper, we first give some examples which belong to the class $\mathcal{S}^*_c$. Also, we show that if $f\in\mathcal{S}^*_c$ then {\rmRe} \{f(z)/z\}>1/2 in $|z|We also prove that if$f\in\mathcal{S}^*_c$and$\alpha\in[0,1)$, then$f$is starlike of order$\alpha$in the disc$|z|<(1-\alpha)/(2-\alpha)$. At the end, we estimate logarithmic coefficients, the initial coefficients and Fekete--Szeg\"{o} problem for functions$f\in \mathcal{S}^*_c$.</div

On p-Valently Meromorphic-Strongly Starlike and Convex Functions

In this paper, we obtain sufficient conditions for analytic function $f(z)$ in the punctured unit disk to be $p$-valently meromorphic-strongly starlike and $p$-valently meromorphic-strongly convex of order $\beta$ and type $\alpha$. Some interesting corollaries of the results presented here are also discussed

Inequalities of harmonic univalent functions with connections of hypergeometric functions

Let SH be the class of functions f = h + (g) over bar that are harmonic univalent and sense-preserving in the open unit disk U = {z : vertical bar z vertical bar < 1} for which f(0) = f'(0) - 1 = 0. In this paper, we introduce and study a subclass H(alpha, beta)of the class SH and the subclass NH(alpha, beta) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H(alpha, beta) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions

Abnormal Intracellular Accumulation and Extracellular Aβ Deposition in Idiopathic and Dup15q11.2-q13 Autism Spectrum Disorders

<div><h3>Background</h3><p>It has been shown that amyloid ß (Aβ), a product of proteolytic cleavage of the amyloid β precursor protein (APP), accumulates in neuronal cytoplasm in non-affected individuals in a cell type–specific amount.</p> <h3>Methodology/Principal Findings</h3><p>In the present study, we found that the percentage of amyloid-positive neurons increases in subjects diagnosed with idiopathic autism and subjects diagnosed with duplication 15q11.2-q13 (dup15) and autism spectrum disorder (ASD). In spite of interindividual differences within each examined group, levels of intraneuronal Aβ load were significantly greater in the dup(15) autism group than in either the control or the idiopathic autism group in 11 of 12 examined regions (p<0.0001 for all comparisons; Kruskall-Wallis test). In eight regions, intraneuronal Aβ load differed significantly between idiopathic autism and control groups (p<0.0001). The intraneuronal Aβ was mainly N-terminally truncated. Increased intraneuronal accumulation of Aβ_17–40/42 in children and adults suggests a life-long enhancement of APP processing with α-secretase in autistic subjects. Aβ accumulation in neuronal endosomes, autophagic vacuoles, Lamp1-positive lysosomes and lipofuscin, as revealed by confocal microscopy, indicates that products of enhanced α-secretase processing accumulate in organelles involved in proteolysis and storage of metabolic remnants. Diffuse plaques containing Aβ_1–40/42 detected in three subjects with ASD, 39 to 52 years of age, suggest that there is an age-associated risk of alterations of APP processing with an intraneuronal accumulation of a short form of Aβ and an extracellular deposition of full-length Aβ in nonfibrillar plaques.</p> <h3>Conclusions/Significance</h3><p>The higher prevalence of excessive Aβ accumulation in neurons in individuals with early onset of intractable seizures, and with a high risk of sudden unexpected death in epilepsy in autistic subjects with dup(15) compared to subjects with idiopathic ASD, supports the concept of mechanistic and functional links between autism, epilepsy and alterations of APP processing leading to neuronal and astrocytic Aβ accumulation and diffuse plaque formation.</p> </div

Intraneuronal Aβ immunoreactivity is not a predictor of brain amyloidosis-β or neurofibrillary degeneration

Amyloid β (Aβ) immunoreactivity in neurons was examined in brains of 32 control subjects, 31 people with Down syndrome, and 36 patients with sporadic Alzheimer’s disease to determine if intraneuronal Aβ immunoreactivity is an early manifestation of Alzheimer-type pathology leading to fibrillar plaque formation and/or neurofibrillary degeneration. The appearance of Aβ immunoreactivity in neurons in infants and stable neuron-type specific Aβ immunoreactivity in a majority of brain structures during late childhood, adulthood, and normal aging does not support this hypothesis. The absence or detection of only traces of reaction with antibodies against 4–13 aa and 8–17 aa of Aβ in neurons indicated that intraneuronal Aβ was mainly a product of α- and γ-secretases (Aβ(17–40/42)). The presence of N-terminally truncated Aβ(17–40) and Aβ(17–42) in the control brains was confirmed by Western blotting and the identity of Aβ(17–40) was confirmed by mass spectrometry. The prevalence of products of α- and γ -secretases in neurons and β- and γ-secretases in plaques argues against major contribution of Aβ-immunopositive material detected in neuronal soma to amyloid deposit in plaques. The strongest intraneuronal Aβ(17–42) immunoreactivity was observed in structures with low susceptibility to fibrillar Aβ deposition, neurofibrillary degeneration, and neuronal loss compared to areas more vulnerable to Alzheimer-type pathology. These observations indicate that the intraneuronal Aβ immunoreactivity detected in this study is not a predictor of brain amyloidosis or neurofibrillary degeneration. The constant level of Aβ immunoreactivity in structures free from neuronal pathology during essentially the entire life span suggests that intraneuronal amino-terminally truncated Aβ represents a product of normal neuronal metabolism